Talk:Ring (mathematics)/Archive 2

4575:

reader who doesn't care will be spared the trouble of skipping over a redundant section." According to that reasoning, this article need not exist - we may merely include a list of links to other articles instead of writing a whole article on rings. Otherwise, your argument implies that only the most important concepts should be discussed and in that case, you are implying that the concept of a non-commutative ring is not interesting enough for this article. I do not wish to pursue this aspect of the dispute further until you explain why the concept of non-commutative rings should not be here (so far, you have argued that it is a burden to read). Thankyou for creating the article on non-commutative rings, since I intended to do so previously but did not have time. In full politeness, however, let me note that the concept of a non-commutative ring is

2749:

and read only the sentence that contains the keyword. Knowledge articles have been optimized for this usage. In other words, we (often) don't employ a textbook format where a motivational example is followed by a formal definition and discussion on how early examples are connected to new ideas. That's a very beautiful format, but that's not how Knowledge articles are usually written because of the aforementioned usage pattern. It is much easier for me to show this than explaining it in words: my draft speaks for itself. In fact, this is related to the lede issue. It is important to state the convention in the intro, since readers (and in fact other contributors) who already know rings probably don't read the definition or examples sections, they would be lost as to what convention is used for the article.

4893:

formalizm. Nevertheless, I have expanded the article based on those IP's changes, by writing two new sections; one on polynomial rings and one on matrix rings. Of course, the subject of polynomial rings is too broad to go into depth in this article. Thus, one can only hope to give a formal definition followed by a few notes on its role in algebraic geometry, commutative algebra, and algebraic number theory. Likewise, the subject of matrix rings plays an important role in linear algebra and noncommutative ring theory, and so we can only hope to give a brief summary of its applications. Does anyone have any particular opinions as to what to include in these two sections? If not, I can proceed at some point in the future to expand these sections along the directions already mentioned. --

4196:

theory in mathematics. As this was not there earlier (except later in the article), I felt inclined to add some points on it. Have you read the whole lede or just the first paragraph? The later paragraphs, I feel, are the main body of what I have added. On the other hand, no one seemed too worried about the article's progress in the past few months - the only changes being minor re-wording and addition of references. Therefore, I feel that the changes that I have made, although not perfect, may constitute some advance towards the article. Rather than criticizing further additions to the article, I feel that we should spend time suggesting further improvements. However, any improvements you can make to my lede are encouraged. --

3046:

Lemma and Hilbert XXX theorems. (Maybe I'm only ignorant, but I don't know any non-commutative counterparts of them.) The article should be about main ideas and procedures (e.g., various constructions) that are applicable to even to non-commutative rings. On the other hand, though my background isn't on non-commutative rings (not even algebra), I believe, for example, localization is still an important tool for non-commutative rings. (e.g., microlocalization) And I don't even have to mention numerous applications to non-commutative geometry. The main focus of the article should be on such topics. In particular, it follows from this view that the article shouldn't contain many examples of commutative cases; they belong to

3571:. I realize (and appreciate) that you've put a lot of work into your version, but that alone doesn't make it worth preserving. You having "kept your cool" is good, I suppose, but I see absolutely nothing that anybody's done here which should cause you to lose your cool in the first place. People changing around and deleting your text is not rude-- it is exactly what happens on Knowledge all the time. Attacking Taku repeatedly for advocating his version (exactly what you are doing for your version) is unlikely to win you any allies. (I'll watch here if you want to reply, but I'm not really interested in being drawn into an argument- consider this my (unsolicited) two cents.) 2489:

clear that the article needed to be reorganized. TM's new organization is not perfect (and leaves out some pretty huge deals from my point of view), but I think it is good for this stage of the article writing. Once we get more of the material added (much of which PST already asked for help adding), the article will probably need another rebalancing. Right now I think the consensus (Salix, myself, and the other leading JS) is that TM's draft should be used for its organization, and that we should merge back in the good content from PST. Once the outline is settled, I'll be more willing to add my content to the article rather than the talk page.

2388:

every reader knows rings, not every reader is non-mathematician either. Every article in Knowledge has to be organized so that readers can find materials on topics that interest them. When reorganizing the article, I was able to see the efforts to increase the accessibility, which is fine, but that shouldn't conflict with the efficiency with the introduction of materials. This is why many math-related articles are accused of being terse, which is justifiably so but you must understand there are good intentions for that format too. While motivations are important, we cannot forget Knowledge articles are meant for reference. --

1988:(after edic conflict) I think I find the structure of TM's version to be a bit better. Bringing the example of the integers to line one should help the layman as an intro with many new technical terms can be intermediating. I do prefer giving the definition first imediately followed by a worked example rather than vica-versa but PST's description of the properties seems to read better. I never like the show/hide button in articles especially when the they are only hiding one or two lines. I agree with Jack that there are good parts in both versions. -- 2011:

some of the emphasis has changed, and as PST put so much work into it, I thought it was a good idea to let him merge back in his own writing. I would not worry too much about the intro, since it will have to be rewritten as the article progresses. It should summarize the article; it is not an introduction to the awe-inspiring majesty that is the mathematical concept of ring, but rather an introduction to this encyclopedia article which is but a tiny drop in the sea of writings about rings. As the article changes, so must the introduction.

4501:

finite groups, these concepts lead to, of course, "the fundamental theorem of finite abelian groups). With regards to "isomorphism", I do not think much more can be said apart from that it is a "homomorphism which is both an epimorphism and a monomorphism". One could of course mention analogues of the isomorphism theorems for groups (or mention Shur's lemma) but I feel that this is redundant. Similarly, the concept of a subring, in my opinion, is not as interesting as the concept of an ideal (although it

4715:

Likewise, the fourth paragraph should be slightly more explicit and also deserves a cleanup. The point in the complexity of the last two paragraphs lies in the fact that this article is not exclusively for "dummies"; that is, it may also be read by professional mathematicians, or perhaps more likely, by undergraduate students in mathematics. However, I do agree that some of the lead can be improved. I will do so as soon as possible and please let me know what you think (a good model is the lead of

1714:

that's how I understand.) In other words, this article shouldn't include the list of definitions of Lie ring, given that this article is about ring in general. (In fact, I just noticed the article doesn't mention noetherian ring, the most important ring!) Since the discussion is too slow, I'm simply going to delete materials that in my opinion are non-essential. (since I'm on Christmas holidays, I should have a plenty of time to devote to this major revision.) --

2847:

understood that a and b are arbitrary.) Second, it would be helpful if you were more specific on which materials you want to recover. For example, I deleted many examples, which can be found in more specific articles (e.g., commutative rings). Since this article is "the overview of rings and the ring theory", specific examples of, say, ideals are not suitable for this article. (Please don't respond to these points here. I want to elaborate on this below.) --

2203:(iii) I'm sorry that if I made you feel like, somehow, I demolished your work. (Sad to say, you just have to be used to that feeling in Knowledge. Natural selection makes Knowledge better.) My intention was to tighten and short the article to create rooms for more discussion on examples, application and such. The article has to include core materials that can be found in any standard algebra textbooks, and the old version, (believe me, which lacks 4960:"ing the set" is a punctuation error, as it requires a comma after "ring". The M-dash after "conditions" should be a colon. The ordinal numbers followed by periods (e.g., "1.") in a running paragraph is a punctuation error: proper punctuation is to surround such list numbers with parentheses when they are within a running paragraph: e.g., "(1)". Ordinal numbers followed by periods is correct when the list is set off from the paragraph, like so: 31: 2507:

meaning of these terms. In general, mathematical terms are written in the section before they are explained. But, the ordering at the moment is low priority. It is more important that we add new content to the article. In particular, Plclark has given some suggestions for improvement and we could probably work on that (if the consensus agrees later, we can change the ordering; there is no point in re-ordering the article every few days).

4282:

see any anything wrong with your edits, I do find that the original lead was better than the current one, in terms of appealing to a wide audience. While we try to make articles as accessible as possible in WP, they are not exclusively for "dummies". Therefore, I feel obliged to revert at least the changes you have made to the lead. If you feel otherwise, we can discuss it here and I will wait for your response before reverting. --

4363:

and a subring are included but there is no comment on ideals. In my opinion, ideals are perhaps the most important concept in ring theory (more important than "zero divisor", certainly). Also, "non-commutative rings" do not merely constitute a "research interest". They are of fundamental importance due to their natural occurence in mathematics. Please understand that although I appreciate your edits, some of them are incorrect. --

2702:

time and effort to do that. Besides, your version has ruthlessly deleted half the content which took a lot of effort to write. More importantly, your version has written this in a grammatically (and mathematically incorrect) way. This is why I suggest that you write from your head rather than copying and pasting from other articles (if you would have done this, I might not have had the same reaction after seeing your changes).

2623:

background. We don't make the intro easier to understand so that a greater number of people can understand it. No, we make the intro as informative as possible to maximize the value of the article for the reference purpose. In other words, the lede should be much more than discussing what a ring is in laymen's terms. Hence, my changes. (In fact, many math-related articles follow this style, I believe.)

4666:

in his bibliography. I don't read any German, so I can't really search for any sources or uses by Hilbert of the term before Zahlkörper being published. Also the relevant part of Cohn's book is viewable on Google books for anyone that doesn't have a copy. I don't have much background in modules, but I think I've distilled what Cohn wrote correctly for what is needed for this article. -

2111:

the theory of their noncommutative counterparts. A fairly recent trend, started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups, attempts to turn the situation around and build the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'.

4539:

that it now exists?). Here it suffices to say that they are more interesting and more difficult, and give a link to that article. The reader who cares has only to click the link; the reader who doesn't care will be spared the trouble of skipping over a redundant section. Not to mention that the editors' work is more effective if each topic is discussed in only one article.

2193:, which states the lead must be the summary of the article, not an introduction to the subject. The old version even doesn't discuss commutative and non-commutative rings or applications in algebraic algebra. (I admit the later is related to my interest.) So, I deleted the mention of the closure property, which is rather a technical detail, and I copied-and-pasted one from 4014: 3994: 3980: 3970: 3946: 3930: 3920: 3896: 3886: 3865: 3851: 3841: 3814: 3800: 4556:

must be defined here only for the sake of the "Alternative definitions" section. Ideals are important enough to deserve a section; but it suffices to give a *brief* definition, a couple of examples (with no discussion), and a paragraph or two, at most, *summarizing* their importance and applications. Anything beyond that belongs to their own article. All the best, --

4530:

it from other related topics. The present version does not define a ring, not even vaguely. After reading two or three screenfuls of dense prose, the reader has still not been given that basic information. He will not be able to tell the difference between a ring and any other algebraic structure. And, therefore, all that prose will not make any sense to him.

4489:

and algebraic geometry which have different goals to ring theory. Therefore, in my view, non-commutative ring theory is more shifted towards the theory of rings (non-commutative division rings alone are complex in nature - there exist non-commutative division rings which are not isomorphic to their opposite ring). Examples in this theory include the famous

4213:

as well as laymen. Over the next month, I intend to substantially change the article, in this respect; while keeping the article accessible, I intend to give a full discussion of ring theory and its applications within mathematics. Currently, the topics included, are not the main crux of ring theory, although they may be simple for laymen to follow. --

1010: 941: 871: 3586:

deleting my text is of course what I want: I want people to improve the article. I was upset with Taku because he deleted half the article without discussion and then re-wrote it in his own words. I will continue to keep my cool and once again I am sorry about getting angry (perhaps you might have felt the same way if you were in my position).

2743:

over again in the past. (For example, the notability policy doesn't dictate what can be included to Knowledge; that's what AfD is for. But with the policy we can save a lot of time not doing actual AfDs.) In other words, citing the policy is never an argument, and, when there is a disagreement, you have to make a case yourself. This leads to my

1819:

extend to complex numbers to allow using more analysis. Actually, just allowing "negative" modules is already helpful, as it is often easier to prove that X-Y is zero than X=Y directly. I think there are lots of other "combinatorial" examples like this (maybe this is called algebraic combinatorics). Probably one could say something about

2690:

with grammatical errors. So, I tend to prioritize the former. I understand that some people have other priorities, but, to me, that just misses the point entirely. GA and FA statuses are meant to give some quantitate information about what is done and what is not done. They are not something to be sought after. (Students study for

1771:

at a Lie group over R, so for instance R could be the integers, the p-adic integers, or various small extensions ("maximal orders" tends to go along with this). For the most part though, Lie rings are considered as Lie algebras over slightly more general rings (some references don't require Lie algebras to be over fields anyways).

3497:

don't have the final say to the format or style of the article or what can be included or not. (You are apparently acting that way.) You are the only who is being too stubborn. Like said before, I'm not planning to replace the article by my version, but you should at least accept the existence of the disagreement. --

4646:. The concern is that active editing in a section of a major page such as this distracts from a tidy article overall. The size of this article was kept down by the fork. But I always expected that an appropriate link to the fork would be maintained, contrary to what happened until the link was replaced today. 4497:. Other important concepts include also, the representation theory of groups and the Brauer group. Therefore, since non-commutative rings constitute a vast area of ring theory, it makes sense to have a detailed discussion of their theory in this article (the current discussion is not even satisfactory). 4837:

Thanks! I do put a lot of thought into this article, but it always helps to have people point out some flaws, as you have done. Of course, the lead can still be improved, but mainly it is the second paragraph which needs improvement in my view. Unfortunately, I have not had much time for this article

4737:

Prompted by the above comments I have gone back to versions of the article which existed late in 2008. The lead there was much shorter and simpler, and I think this made the article less daunting for someone who did not already have knowledge of the subject. I am inclined to agree with Expz: the lead

4694:

Your hard work may be ineffective if your page does not welcome the uninitiated. For us mathematicians, it's too easy to forget how bewildering our writing can sound to the layman. By writing an encyclopedia article, you have the chance to popularize the knowledge of specialists -- don't pass it up!

4555:

Dear "Point-set topologist", to answer your comments: subrings should be properly defined here not because they are important, but only because the reader will not be able to understand other examples and definitions without understanding the definition of subring. Ditto for pseudo-ring: the concept

4529:

Dear "Point-set topologist", i am really disppointed by your wholesale revert of my edits. Can't you really see the problems with your version? For starters, the lead section *must* begin with a definition of the article's topic --- succint, but as accurate as possible, and sufficient to distinguish

4500:

Lastly, I agree with you that there are many concepts in this article which are not interesting enough. It is difficult to decide whether concepts such as "the tensor product of rings" or "the direct product of rings" should be discussed unless they lead to interesting consequences (in the context of

4412:

is 90 kilobites long and is a featured article. Therefore, although some of the material that Jorge has removed was inappropriate for this article (Jorge's removal was appropriate), much more important material is to be added in the future (notable is a more involved discussion of fields). Therefore,

4233:

PST: I had read all of the lead, but I confess I had only skimmed through it: if I find time I will read it more thoughtfully, and see if I can come up with some more constructive comments. Perhaps also I should have made it clear that my comment related only to your latest round of changes: there is

3012:

Great work! Now that all the basic content is added (like morphisms, subrings etc...), we can start giving summaries of the higher level mathematics of rings. What about something on schemes and spectra? I think I could write that a bit later (if no-one does). Also, thanks for re-writing the notes. I

2764:

presents the topic in an easy and accessible manner, well written, and provides information. As you say, you are not interested in the FA procedures. Well I am and that was my goal when I improved these articles. If you don't like FA procedures, I suggest that you improve other articles which are not

2748:

Forth point. Knowledge being a reference work, its articles are meant to be used for the reference purpose. It follows that, for one thing, you shouldn't expect your readers to read the article from top to bottom. You can't even expect they read a section thoroughly. Some simply search for a keyword,

2689:

Second, about GA. Why hurry? I think it is important to focus on the long-term goals (i.e., increasing and improving the coverage of math-related topics) than short-term ones (i.e., GA or FA status). Or, at least, GA or FA statuses are just not my priorities. To me, informative articles beat articles

2565:

The other thing you said was that the choice of version depends on the person. It doesn't. It only depends on the Wiki policies. Some criticizm of your version is that you use "we" too often, your version is not accessible (i.e don't use mathematical terms before you actually explain their meaning; I

2420:

But that narrow "community" doesn't include me, and doesn't represent the whole community of Knowledge. Anyway, I have to be more specific. What I meant to say is that that article is a bad example "for this article". Since a ring has much more complex structure than a group, it makes sense to employ

2010:

Yes, let me emphasize: there are good parts of both versions. I like the structure of TM's version better (and I think that was more or less his main goal), and that using his structure we should merge back in the good parts from PST's version into their new places. It's not a simple copy-paste, as

1958:

I spend a lot of hard work to improve the article. Of course I don't own it, but removing almost everything I wrote (some 500 edits) in 10 edits is not something that should be taken lightly. I suggest that we keep the current lede (which beautifully explains the concept of a ring to someone who does

1952:

Grammar: I certainly respect the fact that you know more than one language (I myself know more than one language and therefore I know how hard it is to learn another language). However, sometimes, your edits are not gramatically correct (mathematically correct of course!). This means that we (I) have

1862:

Howdy, I noticed the big changes and the two reverts. For the most part Taku's changes looked good, but it did delete an awful lot of material. Unfortunately the revert also deleted Taku's new material. The revert-revert again deleted some of the old good material. I think it would be a good idea

1770:

BTW Lie rings get a lot of use in group theory. The idea is to take the success of Lie algebras over fields, and allow it to work over fairly tame rings like Z/nZ, for n not prime ("Lazard correspondence" is one name for this). In general, it is useful to look at a Lie ring over R whenever you look

1648:

is (if you do, ignore this)? I didn't even learn that when I knew algebraic topology quite well but yet it is a crucial object in algebraic geometry. I am just saying that there are important concepts in mathematics (that some people do care about) that most of the population does not know (Lie rings

4665:

I've added a bit from a source I found, though it could use some review and adjustment perhaps. Cohn notes Hilbert introduced the term in 1892, so that's good enough to quote that, but Cohn doesn't list any publication of Hilbert's but the 1897 publication of Die Theorie der algebraischen Zahlkörper

4574:

To begin, you have written, "Non-commutative rings, for example, shold be discussed in their article (did you notice that it now exists?). Here it suffices to say that they are more interesting and more difficult, and give a link to that article. The reader who cares has only to click the link; the

4570:

Sorry for the delay in responding - I did not see your comment until about a few minutes ago. Firstly, I did not delibrately revert your edits in their entirity (a simple undo button would have done that, but notice how several edits on my part have occurred after your edits: not just one "undo"). I

4452:

As for the "non-commutative" section, it was moved to a separate article, where one can discuss with more depth and detail. We could copy back some of that material to this article; but, in my opinion, only if it connects to other parts of this article. (For example, subrings and isomorphisms must

4362:

Let me make it clear that I appreciate your interest in the article. I also agree that many of your changes were appropriate and so should remain. However, I intend to correct some minor points in your changes. For instance, under the section "some important concepts", the concepts of a homomorphism

4330:

I have no problem with adding a bit more detail to the lead, such as a note about non-comm. rings being a topic of research. However, there is already a section where the research topics are dicussed in full, and any longer discussion should go there. The lead is not meant to say *everything* that

4281:

I certainly agree that the article was too long. However, the lead of an article is aimed for the widest possible audience. Therefore, although we aim to explain the concept of a ring to laymen, we also aim to describe how the notion of a ring applies to other areas of mathematics. Although I do not

4263:

Repeat, WP is not a textbook. Yes, articles should try to be accessible to the general public; but they should also be as concise as possible, as befits a reference work. And they are not supposed to be self-contained: that is what wikilinks are for. There is no point, for example, in explaining

4190:

Hi James. Thankyou for taking interest in the article. I feel that, as an encyclopedia, we must make this article relevant to both laymen and mathematicians. When I first become a user of Knowledge, this article was essentially a stub. The previous version, as you point out, consisted mainly of what

2742:

Thirdly, about policies. I don't think you understand how Knowledge works. Policies and guidelines are meant to make contributors productive (not put constrains to them). The point is that, without codified sets of rules, contributors have to spend too much time on matters that have come up over and

2701:

You have not understood me. Before I made 500 or so edits to this article, the version was at the same quality as your version; grammatical mistakes, mathematical mistakes and formatting errors etc... To get your version back to good quality, would take at least a few hundred edits. I don't have the

2573:

I want to put this in the politest possible way, but your version is start class. It requires significant revision (which I can't do all over again (it took a lot of hard work)) to make good. Technically, the other Wikipedians say that your ordering is better. I would appreciate it if you understood

2557:

The other problems with your version is that you seem to have blindly copied text from other articles. While this is not against any Knowledge policies, there are many errors in those articles. So I suggest that in future, before copying and pasting, check for errors first (and this will also ensure

2292:

I don't care much about the lead. Do you think you can accept the "basic structure" of my version? If not, we have to submit ourselves to the community and let them decide which version they like. Like I said above, the structure of the old article is unacceptable to me (whence, the change). Nothing

5046:

I agree. Perhaps this started as a claim about the inverse image of 0S which does hold in rings without 1 (which some people call rings), and was later "corrected" to the current form? I have marked the sentence as "dubious" for the moment, but since it's false it should obviously be removed rather

4753:

Since we have all agreed that the lead should be shorter, I shall rewrite the lead tomorrow and shall request opinions. Please give me this time to do so. You may revise the new lead once it is finished (although you are, of course, entitled to edit the lead in its current state, I would appreciate

4710:

I should note that the current lead is due to me. The point is that I wished to include some of the various applications of ring theory in other branches of mathematics, as well as ring theory in its own right. If you look through the history of this article, you will find various other versions of

4680:

I know a number of you have worked hard on this page, especially to include nontrivial details and lots of useful information. However, the introduction to this page is both intimidating and unusually long. I suggest most of the introduction information elsewhere, and leaving the intro to emphasize

4538:

Finally, unlike a textbook or popular science article, a WP article is not supposed to be self-contained and exhaustive. On the contrary, one should avoid as much as possible repeating text from other articles. Non-commutative rings, for example, shold be discussed in their article (did you notice

4212:

Let me also add that clarity of an article, and the extent to which it is straightfoward are not the only aspects that are to be taken into account. For instance, there are many students who may wish to learn the applications of ring theory within mathematics. We must appeal to those students also,

3511:

I accept the disagreement and have accordingly promised to add parts of your version back. The point is that you say that you don't want your version, then that you do etc... I have explained why your lede is not appropriate and you have not responded. I appreciate your efforts in this, but I would

2774:

I also intended to add more information which I would have done instead of wasting my time typing here. My point is that your version was done without discussion, with your purpose (not to make the article FA, but make it back to how you like it), and in your style (grammatically and mathematically

2670:

your lede (which is a good thing). It is copied and pasted from another article (start class) and hence the errors. I suggest, that if you really want your lede, add it to beginning of the "examples and applications" section. But first, I strongly suggest that you rewrite it in your own words (with

2200:(ii) Grammar. Yes, that's a problem :) I was interested in the organization, and so my writing was rather rushed. (I thought there is a plenty of time to polish it later, so I didn't mind. I intended to go over later to fix syntactical errors. or maybe I can't and need others' help on this, though.) 2134:

The ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its

1942:

Lede: The lede has to give a basic overview of the subject. Now, you obviously know that a binary operation combines two elements to form a third but does the general population know this? Would the general population know that addition and multiplication are binary operations and not just addition

340:

does not seem to have anything on the center of a group and from what I read, it looks like a top-standard article. So should this article contain anything about the center? Anyway, I think that ideals, quotients, subrings, morphisms and direct products are most important (and basic) in ring theory

5003:

In the article's source code, the ordinal numbers followed by periods are formatted as a set-off list with single line breaks, so in the code it looks like a set-off list, but it doesn't display as a set-off list. So perhaps whoever wrote this list in this manner was confused as to how it would be

4488:

Let me explain my concerns for removing the section on "non-commutative rings". Analogous to finite group theory (where non-abelian finite groups are the most interesting), non-commutative ring theory is also very interesting. Commutative ring theory leads to fields such as algebraic number theory

4195:

but of course material was added by many other people also). Although the opening paragraph of this lede may seem somewhat strange, I intended it only to be a "draft". The other paragraphs were essentially non-existent prior to my changes, but are important as they explain the significance of ring

2861:

How many times do I have to tell you? I have politely discussed with you, agreed to add your changes and yet you persistently insist that your version should be chosen. You say that Knowledge articles are work in progress. What you fail to understand is that your version will take at least another

2630:

Yes, but is your lede a summary of the article? Your lede comprises of two opening sentences: both mathematically and grammatically incorrect. My lede explains those first two sentences in depth. Second of all, the remaining part of your lede does not at all summarize the article. It just explains

2488:

I think using the group (mathematics) outline was a very wise thing to do while PST was adding all that new content. Many wikipedia articles have basically no organization at all, and using an FA as a model is a very good idea. However, now that PST has added all this good content, it has become

2110:

Due to its intimate connections with algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, their theory, which is considered to be part of commutative algebra and field theory rather than of general ring theory, is quite different in flavour from

1818:

are reasonable examples of this. From my point of view, you have these modules that you declare to form a basis of the ring, and use direct sum and tensor product as the ring operations. You can extend the "scalars" to include rational numbers to let you only take "half" of a module, and you can

1713:

I should have been more specific. (Sometimes, when you are in a grad school, you get the impression that everyone knows Lie algebras.) Yes, my point was the term "Lie algebra" is probably much more known, and it probably makes more sense to discuss Lie ring as one form of generalization. (At least

4535:

Finally, an encyclopedia article should not be written as a textbook. This is not the place to give proofs of theorems, derivations of formulas, or explanations for straightforward examples. Just saying that the integers are a ring is more than enough. Moreover, unlike a textbook, examples (when

3496:

the disagreement. Why is that so hard for you to accept that someone might not agree with you? (So, he or she is going to change an article accordingly.) Like said above, that's how Knowledge works; either you accept this nature of the editing model or you must leave. This isn't your article; you

3045:

I strongly disagree. This article shouldn't be just about "introduction to ring theory", it should be much more. By this I don't mean to suggest the article discuss ringed spaces or, even more generally, scheme theory, because they're topics in commutative algebra. Ditto to things like Nakayama's

2947:

I have decided to spruce up the article by adding some higher-level content. I made a new section with the above title. The idea of this section is to discuss some of the many ways that one can (functorially, although how explicit this should be is as yet unclear to me) associate a ring to some

221:

Adding images is not the most important point at this stage, I think. What I can recommend as a working procedure: figure out what you want to be in the article, put the links in the "see also" section and start writing on the more important items in the see also section. Also, at an article like

202:

You can move an image by moving it to the line (in the source code) where you want it to be. However, remember that a commutative diagram and tensor products will not be helpful to readers not acquainted to this (and adds, IMO, nothing to the understanding conveyed by the equations). Also, giving

3648:

I just want to point out that I am sorry for getting upset in the end of the last dispute. In the beginning, I acknowledged Taku's mathematical knowledge as well as his work. But it got to the point where he just ignored my comments. I think that the dispute is now settled and we can get back to

2387:

article. (I personally see it is a bad example.) So, some explanation on my intention behind the radical changes might help. Knowledge being a reference work, it is important to get to the point as quickly as possible. This is why I moved the definition above some motivational example. While not

2561:

Also, as you confirmed, you intended to fix up your version for grammatical mistakes later. How much later? Fixing up those grammatical (and mathematical errors) would take months unless you want to work the whole day (for 5 days like I have done). I intend to get this article to GA as soon as

2506:

I explained why I don't agree with TM's ordering on your talk page, but my computer has serious problems and this did not get saved. But mainly, TM's ordering was not accessible; it said in the lede that "all rings in this article are assumed to be unital and associative" before explaining the

4582:

With regards to the lead, I do not see why the definition should be included at the start. Leads in Knowledge articles are designed to summarize various important aspects of ring theory. The formal definition has its place right after the lead, and the reader need only scroll down to read it.

3585:

I will chill out. I did get a bit upset but the fact is that I explain what is wrong with his version, he does not respond to this, and again says his version is good. This takes up a lot of time for me. I am sorry for getting upset and I guess this was unnecessary. People changing around and

4714:

Of the current lead, the first paragraph should be easy to understand. The second paragraph should also be relatively easy to understand for some people with little background in number theory. The third paragraph is, except for the first few lines, quite complicated and deserves a re-write.

4260:

is not encyclopedic. It is too verbose (over 70KB!), with explanations that would be excessive even in a textbook -- which WP is *not* meant to be. It discusses at length topics that are (or should be) discussed in separate articles. The lead section is overly long; while it pretends to be

2846:

you are with grammatical errors; they are easily correctable. Remember articles in Knowledge are work-in-progress. Maybe when you write, you make no grammatical mistakes; I do and asking me not to do isn't going to work. Coincidentally, when you write "rings with the property ab = ba", it is

1774:

I agree that mentioning connections between Lie algebras and rings is appropriate in this article, but that there is no need to specifically go into Lie rings. I would prefer that both ring to Lie algebra and Lie algebra to ring functors were mentioned, as the current (as of a few days ago)

4892:

It was really about time that we had a few sections on polynomial rings and matrix rings, and thus it was great to see the recent improvements in that direction by an IP. I was however slightly skeptical about those changes because they did have a few technical errors; a result of a lack of

2549:

Taku, I would like to emphasize the comparison of the two ledes. First of all, my lede is a summary of the article; it discusses (in simple terms) what a ring is. Your lede on the other hand, is definitely not a summary of the article. It gives the applications of rings to other branches of

5030:

In the section on homomorphisms, the article states that "If f is a ring homomorphism from (R, +, ·) to (S, ‡, *), the inverse image of the identity element 1S of ‡ (that is, all elements of R that are mapped to 1S by f) is a subring of (R, +, ·)." Is that really so? Take for example the

2622:

First, about the lede. I believe the lede should not only introduce a subject but is the summary of the article. From this follows inevitably that the lede may contain many technical terms and mentions of important examples and applications, which may require the readers to have some prior

4345:

Please discuss every change you make. Your recent deletion of the section on "non-commutative rings" was inappropriate. Although I do not think you have bad intentions, I should politely request that you explain dubious changes for otherwise there is no reason why I should not revert.

3126:

Gives an example of the integers too quickly. The first sentence should start by writing about binary operations. It is not clear to the reader yet whether addition is a binary operation or not (if someone reads this article, he will know very little about rings (if at all anything)).

1553:

More inline citations (I don't have access to a library for sometime so if you could add even one citation I would be grateful). Especially, if anyone has the book by Lang, could they find the page numbers corresponding to certain facts in this article (Lang is the most cited book on

5104:. However, since the article mainly uses the more restricted concept of ring with a 1, the statement should be either removed or else rephrased to indicate it is using the broader concept of ring. On the whole I think it is a fairly unimportant detail anyway, and I shall remove it. 2656:

Also, I don't understand how you got the idea that my lede is in laymen's terms. To write a good lede, one must make it as accessible as possible as well as mathematically precise. My lede gives a mathematical term, then explains it. Don't you think that this is better than writing

457:

information. For example the proof of 0a = a0 = 0 is certainly not encyclopedic. If you absolutely want the reader to know about this, provide a precise reference to a book, including page number or chapter. This way, the information will be available, but does not clutter up the

570:

Provide more orientation to the reader, i.e. what are the main classes or types of rings (e.g. comm. / non-comm)? The examples list is pretty long, try to give it more structure by explaining the relations between the examples. When explaining the basic notions, come back to the

1878:

Yes, sorry about the radical changes. I went more than I initially planned. I deleted lots of materials since I was trying to create a better organization. After the reorganization, I thought we can just readd old materials back. Again, I should have been more clear on this. --

1863:

to try and merge back the good parts of PST's material deleted by Taku in small changes with edit summaries. For instance, at first glance it appears that Taku deleted some important references that should probably be merged back in somehow (but not by reverting his changes).

3084:

Therefore, I am removing your link (you specifically stated that you agree with my version), and am going to edit the article. If you want to make an article in your style, I suggest you do it to another article or use the sandbox. As I said, go ahead and make your changes to

3623:

I'm sorry, but what the hell is going on with the page now? There's a mammoth amount of talk about the article pretty much just from two editors i.e. a dispute... is that resolved now (as the below section indicates? If so, what was the resolution? The whole debacle makes it

2800:

surprised by how my edit upset you, for that, I'm very sorry. I don't even insist that we use my version, since now I know how this article meant for you and my version doesn't mean (personally) anything to me. One good editor is much more valuable than one good article.) --

4532:

Moreover, the lead section is not the place to go into lengthy explanations as to why rings were invented. At most, there should be one paragraph mentioning the topic's importance and the general are of applications. Detailed discussions belong in the article body, if at

4059:

I always like to know the rationale for mathematical terms; it helps me feel like I know what I'm talking about. What did Hilbert have in mind when he called these things "rings"? I can see why someone would call SO(2) or even SO(3) a ring, but those are groups, not rings.

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I do not agree that the original lead was better for the lay reader. For one thing, it did not really define a ring, not even vaguely; so all the discussion about the merits and applications of rings will be incomprehensible to readers who do not already know what a ring

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Interesting (which is a standard response when you don't know what is being discussed.) Yes, a cohomological point view might be interesting to discuss in the article. Since my expertise is limited to commutative rings and algebras, I'm not a good writer for that task. --

2157:

This is why I think my lede is better. If necessary, I can critize the other content too (there is of course good content and I will add that back when I next get the time to do so). I think with such dramatic changes, it is always good to explain first and then change.

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is important; it should only give a *short* overview of the article's topic. Also, any technical discussions about the properties of non-commutative rings (or any special kind of ring) should be moved, whenever possible, to the corresponding article. All the best, --

463:

Do you by chance know any good books on ring theory? I don't have access to a library in the near future so it will be difficult for me to do that. If you know any good books for citations (ref section is perfect), I would greatly appreciate it if you added those.

4379:

Another example: it is written that "the integer elements of any ring with unity element is isomorphic to the ring of integers". What if the ring is finite or has non-zero characteristic (the statement is almost correct but too basic to be included in its correct

1790:

Thank for enlightening me. The note on a Lie ring would be much nicer than simply the list of definitions. The current article (new or old) is short of discussion on the use or applications of rings. (And covers too much on elementary examples and properties.) --

2723:

and such. If you feel that this article is not informative, go ahead and add information. What I was unhappy about was the fact that you deleted information rather than adding it. This clearly contradicts your claim that you want the article to be informative.

344:

One more question: I am thinking of adding an example of a finite ring. The most obvious one would be of course the the integers mod n with its natural operations but I am not sure how to get an image of a 'ring table' for this. I noticed that there was one on

837:

Wouldn't it be simpler to just say it's the cartesian product of the rings in question equipped with the pointwise operations? I think for this section we can assume the reader either knows what the cartesian product is or is willing to click on the link.

3073:

a) There are articles on every concept in ring theory. If there is already an article, why add more here? This should give a basic background and then link to the articles which explain the "advanced concepts" in depth. Plclark has done this and I agree.

2815:

As I said, I will add the good parts of your version; I won't add back what you have re-written nor will I delete the content that you have deleted. What was most upsetting to me (I was not actually upset) was that you made these deletions and re-writes

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important (IMO) than giving every little detail of all the definitions. This article is to give an overview of related notions. Hence it may have to be a bit short at times, if necessary even sloppy (but make clear what is sloppy if you need to be

3472:

I don't want to waste anymore time here. Again, I request you to edit another article; if not, I will not respond to anymore of your comments (sorry but I have been as polite as possible and yet to continue to behave in this (impolite) manner). I

2263:

But that doesn't work for the reasons spelled out above; whence, the reorganization. (I'm really not an expert on algebra since my background is an analysis.) Do you think you can work with my version? by fixing errors and polishing the text? --

1898:(to check the style of the article). The article should be accessible (don't use complex terms in the lede). Please have a look at the link and compare both articles. This should explain why your change weren't completely appropriate. Please ask 3186:

This is completely incorrect. For a start, the sentence sounds like "a ring is an Abelian group with multiplication". No. It is an Abelian group under addition. Even then, it is not clearly stated what a ring is under multiplication (a monoid).

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identity which was required. In the general notion of "ring" (not required to have a 1) the statement in the article is perfectly correct, but expressed in a misleading way. If the statement is to stay then the notation should be changed from

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the mathematical term. Furthermore, even though my lede does not use complex mathematical terms, at least it is correct, yes? Your lede on the other hand is both mathematically and grammatically incorrect and is better suited to the article,

2048:

I don't think this sentece is well-written. Notice: "A ring is an algebraic structures". More than the extra "s" at the end of "structure" (grammar), how many people would know what a algebraic structure is? My lede explained this clearly.

4298:

Let me give one example. In the old lead, there was some description as to why non-commutative ring theory is almost as active an area of research as commutative ring theory. The current lead fails to note this. To prove my assertion, see

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500 edits to improve. I have already worked hard on this version and yet you want to undo it with 10 edits. Please stop discussing here and edit another article. If you do not wish to do this, I won't respond to any further comments. --

5007:

The semicolon within "under multiplication; such that multiplication" is incorrect, as the "such that multiplication ..." is not an independent clause. Instead, a comma is required here in order for the current phrasing to be correct.

2948:

other kind of mathematical object. I think this ia good way to see that rings are interesting and useful in many different areas of mathematics. Here are some ideas for such things: others should, as always, feel free to add more:

1697:

Anyway, given that we have an article which is not yet scratching the surface of the enormous theory of rings -- i.e., with an associative multiplication -- it is an easy choice to exclude consideration of non-associative algebras.

2225:

Thanks for the understanding. I know that you are certainly an expert on the subject. More important than re-organization would be if you perhaps write about the important concepts on the topic and then add it to the article.

3512:

like to request you to allow us to get this verson to FA. So equivalently, I am saying that I won't respond to anymore of your comments: I can't prevent you from writing stuff on the talk page but at least I can do this. --

1953:

to go through some of your edits which is a (minor) problem. I don't want to be rude; you write English fluently but there is always the minor grammar problem now and then. Please see the criteria for featured articles.

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homomorphism from Z to Z_4 (mentioned in the previous line of the article). The inverse image of the multiplicative identity of Z_4 is {...,-3,1,5,9,...}, which is not closed under addition, and thus cannot be a ring.

4136:

The sentence "The term rng (jocular; ring without the multiplicative identity) is also used for such rings." should be placed after the description of rings without a unit, not after the description of unitary rings.

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incorrect). I don't mean to sound rude but I have kept calm since I saw your changes. I approached it in a logical manner (rather than getting angry). Before my changes, the article was start class. The length of the

3214:

is not a ring unless you impose the necessary binary operations. Second of all, "the space of continuous functions", has no meaning because you are not specifying their domain and range (the range has to be a

1688:(who has a master's degree in mathematics) has a wealth of links on his userpage, including a whole list devoted to Lie algebras and several links to algebraic geometry topics. What he was saying is that the 2550:

mathematics. If you want to add back your lede, I advise you to add it to the "examples and applications" section (and again I would like to request you to read my criticizm of your lede on the talk page of

4108:

within the ring, and the emptyness then suggests the solid torus or anchor ring kind of figure, where the whole is formed around a certain gap. This figure contrasts with a field where a mere point marks

2615:) but just corrections and such. I have already explained what is wrong with Taku's version (except for the order of material, I think most people agree that his version is not appropriate, right?). 3668:

There needs to be a section on the field of fractions. If anyone has the time, could they please create it (my hands are full at the moment but if no one creates the section, I will do it myself). --

2888:), and probably won't make it through the GA/FA filter. That image could easily be duplicated as a wiki table, and perhaps a better (pictorial) image could be found (though I don't have any ideas). 226:, otherwise you end up deleting lot of content you wrote earlier. For example, the subring section will be too long when the article reaches maturity. Finally, it is also necessary to remember that 3206:, the other being a group. Examples of rings include: the set of all integers, the space of continuous functions, polynomial rings and the ring of matrices (but not the set of traceless matrices). 3567:? To suggest that Taku is simply wasting his time and others' is fairly ridiculous. To ask him to stop contributing to this article and discussion is also inappropriate. You may want to check out 4815:

Better there are people editing the article who've given it serious thought than everyone editing willy-nilly. Knowledge always welcomes contributions, but what good to anyone are careless ones?

1754:. Since groups and rings are distinct concepts and not even remotely similar (similarity in definitions doesn't correspond to that in properties or behaviors as we all know), I don't see why. -- 3077:

b) Secondly, I was not rude to remove your link. I was as polite as possible when I saw your changes not to mention the fact that I kept my cool. If you like your version, go ahead and do it to

2516:

Other problems with TM's version: Use of "we", grammar, mathematical precision, accessibility etc... (One reason for these problems is that he copied a lot of material from other Wiki stubs).

4413:

the article's size may well reach 90 kilobites or so in the future. Many of Jorge's changes were appropriate but some were not (in my opininion). Therefore, I also request that Jorge explains

4076: 3013:

think it now gives some useful information to the intermediate reader. I will patch that section up (in terms of linking, for instance I don't think that equivalence relation is linked) now.

1611:

The article is a bit too long despite that it doesn't cover many important topics. I propose we delete many discussion on elementary properties and some specific examples. For example, is a

4505:

interesting). More emphasis should be given to ideals before a full discussion of subrings is commenced. If you feel otherwise on any point I have mentioned, please feel free to comment. --

3316:

rather than of general ring theory, is quite different in flavour from the theory of their noncommutative counterparts. A fairly recent trend, started in the 1980s with the development of

1969:

These were the main problems with (most of) your edits. I will try to incorporate them into the current article but for now, I have to go. Nevertheless, I appreciate your efforts in this.

110:. For example, a few hours ago, it contained nothing about morphisms of rings! I have made some improvements to the article and hope to get it to a higher standard; perhaps something like 4408:

I have a few more comments to make. First and foremost, let me note that the article's size has now been reduced to 61 kilobites (a 9 kilobite improvement). On the other hand, note that

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was five times the length of the article. What I thought was: why are people wasting their time talking instead of implementing? But again, it seems that we have gone back to talking. --

4434:

Correction: the initial size of the article was 76 kilobites, so a 15 kilobite improvement is quite reasonable. This would not be without Jorge's input so let me thank him for that. --

4571:

attempted to revert those changes which I felt were inappropriate. That said, not all of your changes were inappropriate. I will now attempt to respond to each comment you have made.

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for WP articles linking to this one. This might give you some ideas on what to include. Other than that, looking into books, scientific articles, talking to people will give more.

2068:

This sounds as if "a ring is an Abelian group with multiplication"; it is under addition. The c(a+b) = ca + cb is not TeXed properly. "Technical" should be "mathematical" etc...

554:

This I definitely need to do (like one interesting (but simple) problem is to determine under what conditions a quotient ring of a ring is a field; this could be a motivation).

1667:

of examples. Remember, that not everyone is as clever as you think they are (read the article as if you have never heard of rings before; would you be able to understand it?).

2796:(P.S. I'm not personally offended at all if that's what worries you. It's been long since I started the contribution to Knowledge, so I'm used to the dispute. Actually, I was 336:

Thanks Jakob; I fixed that up. I will do the same for the image when I get more time. I am just not really sure which concepts should be included in the article. For example,

3089:

and see what other people say. Please stop wasting time here (and other's too; we want to expand the article, not gabble on here as people have done for the past 6 years). --

4261:

addressed to the layman, it never explains what a ring is, and throws a lot of specailized terms that even a mathematically educated reader cannot be expected to understand.

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be defined here because they are used in other important sections, especially in examples; while a single line and a link is probably sufficient for, say, tensor products.)

2405:

has been accepted (by the community) as one of the best articles in Knowledge (that is why it is FA). If you think otherwise, the community, probably won’t agree with you.

2041:

and I was aiming to have exactly the same structure as that. Anyway, I think that the current lede should be kept. I will compare and explain (the following is TM's lead):

544:

Motivate and explain more what is going on. (The motivation section is well-done in this respect). For example: what is the purpose of the quotient ring? (This is actually

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On the 20th of June user Charvest started a section on finite rings that I expanded. A month later it seemed this section was due its own page, so I moved the section to

3081:. I think that if you do, God knows how many people will revert your changes. In this case, people don't say anything because it is not they who have done the hard work. 2189:(i) the lead. This is a delicate issue, since it's about the personal preference. I can see why you prefer your version to mime. I personally try to follow the advice at 142:

You are absolutely right. Many of the (even top-importance) algebra articles contain little more than the mere definition. So, simply go ahead! Beware that there is also

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at the reference desk a while ago, but the only definite answer was from a very dubious source (at least one of the etymologies given on that page is totally wrong).

3267: 2197:. But like I said it's more about personal tastes, and I'm happy leave the community to decide what type of the lead it wants. (My position on this should be clear.) 2044:

In mathematics, a ring is an algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.

2366:

I don't agree with the structure. As I say, please consult Jakob (an expert in writing Wiki articles). I don't really think that your changes were for the better.

1684:

I fear that after less than a week in his new incarnation, Topo's civility may already be degrading. Topo, if you had taken the time to check, you would see that

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is supposed to cover a more "advanced" treatment of rings. This article on the other hand is supposed to cover the more basic concepts of ring theory (compare to

1810:) is a cool use. Another nice use of rings is: when trying to understand some set of examples, you form a ring generated by that set and study it instead. The 4583:

Nevertheless, I have now added a link to the formal definition at the end of the first sentence. Should the reader wish to see it, he need only click the link.

3414:, meaning number ring. Although the first definition of rings was given by Fraenkel (1914), it was Hilbert who popularized the term. The term also appears in 3457:

You have not explained what unital and associative mean. This should be given after the formal definition (so you have not ordered the article correctly). --

3611: 3324:, attempts to turn the situation around and build the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of 3029: 2711:

Also, I understand that you wanted this article to be informative, yes? But what was uninformative about my article? Remember, this article is about the

2596: 2532: 2455: 414: 387: 130: 1966:"with the property that ab = ba": for which a and b? All a and b in the ring? Mathematical precision is very important especially in featured articles. 1644:

etc... are very important in mathematics. Just because you don't know something doesn't mean that it is not important. For example, do you know what an

2315:; it is important that you not only contribute content but make sure that you don't overlink, use "we" (unencyclopedic), and destroy structure etc... 4455:

As for article size, I agree that it is not a problem per se; there is much material that could be added. (For example, what do you think of merging

1959:

not know anything about it) compared to your lede which requires some depth of knowledge to comprehend. One other problem with your lede (you wrote):

1921:. Could you explain why you are unhappy with the changes. Is it only because the style of the new version differs from that of the group article? -- 1572:

The "Examples and applications" section has to be expanded; particularly a summary (like group (mathematics)) is necessary at the top of the section.

3054:(should we create one?). (I guess this explains why Point-set topologist can't accept my proposal for the structural changes, at least partly.) -- 2278:

I will look at them now. But as I explained I think that my lede is better (so I may not add back your lede but I will add back your other edits).

3219:). I don't know head or tail of what you mean by the "ring of matrices" (mathematical precision again, you have to specify the field etc...). -- 2185:

Thank you very much for responding to my post. Now I can understand what made you unhappy, (and I'm very happy that I now know it.) To respond,

187:. Could someone please position the image so that it is a bit lower (and is on the same row as the axiom for associativity of multiplication). 1107:). The operations are addition and multiplication of functions. More generally, the set of all continuous functions from a topological space, 430:

As per p.s.t.'s request, I give some points which I think would improve the article further (currently, I think, the article is Start class).

3753: 1328:

Some of these examples need to be re-written into their own section. In particular, I think that the last two should be deleted altogether.

4485:). That said, it is practically impossible to cover every concept in ring theory; some concepts must be given more priority than others. 4167:. Together these have substantially transformed the article, without any discussion or opinions about the specific changes made. I find 4144: 2694:, but ideally students should study for the sake of the subject itself and use exams to help them access their degree of understanding.) 146:(which is in slightly better shape, after some recent edits of mine), so it is important to have a clear picture of what belongs where. 5012: 4806:

I like the draft much better! Very welcoming for the new-comer, yet still with useful leads to the one looking for deeper information.

3778: 3772: 3749: 2337:) and then explain how this generalizes. As I said, Jakob is an expert on FA's and GA's (and if he agrees with your structure, fine). 2037:

Certainly. TM's version did have some extra information which should be added back. On the other hand, I know that Jack has worked on

4685:

1. Rings are a simple concept: a set with two operations, one like addition, the other like multiplication, perhaps without inverses.

4450:

First, my apologies for the errors that I introduced. I will fix those that I can recognize, ad ask that you help me find the others.

2138:

Throughout the article, a ring is assumed to be unital and associative. See also glossary of ring theory for additional terminology.

2087:

It should be, "Commutative rings (a ring with the property ab = ba), are much etc... Even that is wrong because ab = ba is not TeXed

851:

I started editing the article). That is exactly why I kept a cleanup tag (per your reason). I will rewrite it and add it back later.

76: 71: 59: 3375:), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as 4493:(first noted in the famous paper by Nathan Jacobson - "On the structure theory of rings without finiteness assumptions") and the 3986: 3734: 1732:. Could you explain what's wrong with it, Point-set topologist? It needs more polishing, but to me it is in a good direction. -- 593:

The article needs references. For the moment, if every bigger section has one, this would be good. I personally recommend using

2142:

What does 'unital' and 'associative' mean (people may not know)? This was (appropriately) placed in "notes on the definition".

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In technical terms, a ring is an abelian group with multiplication that distributes over addition (i.e., c(a + b) = ca + cb.)

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Looking over your changes, I must say that some of the content you have added is important. However, I encourage you to read

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identity. Thus where Gabn1 wrote "the inverse image of the multiplicative identity of Z_4", it was the inverse image of the

3281:

are in the ring. I have no idea head or tail of why commutative rings are much better understood that non-commutative ones (

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2. Rings are ubiquitous: integers, real numbers, sets of polynomials, sets of functions, suitably defined, all form rings.

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no doubt at all that, if I consider the total effect of all the work you have done then it is an excellent piece of work.

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To qualify as a ring the set, together with its two operations, must satisfy certain conditions—namely, the set must be

3684:

Good idea, in fact would it even justify an article? I will think about possibly writing it, whether section or article.

4964:

To qualify as a ring, the set—together with its two operations—must satisfy certain conditions: namely, the set must be

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is a featured article that has similar structure ("textbook examples" and "introduction") to this article. Regards, --

2383:

This might be to beat the dead horse, but about FA and group, I don't see why we have to follow the structure of the

1324:

j , j = +1. A ring analogous to the ordinary complex plane but substitutes conjugate hyperbolas for the unit circle.

3387:"The ring theory". What sort of language is that? This article is about rings, not ring theory. Write this lede at 3313: 38: 4894: 4839: 4787: 4755: 4720: 4611: 4595: 4506: 4435: 4418: 4384: 4364: 4347: 4312: 4283: 4214: 4197: 4164: 4039: 4023: 3806: 3669: 3650: 3603: 3513: 3478: 3458: 3427: 3392: 3337: 3286: 3220: 3188: 3128: 3090: 3021: 2929: 2863: 2824: 2780: 2725: 2672: 2588: 2524: 2447: 2406: 2367: 2338: 2316: 2279: 2250: 2232: 2172: 2159: 2143: 2120: 2096: 2069: 2050: 1972: 1903: 1747: 1670: 1597: 1531: 1510: 1329: 1075: 1022: 953: 883: 852: 640: 630: 612: 555: 529: 465: 439: 406: 379: 355: 326: 212: 188: 170: 151: 122: 368:

I was just wondering: I am not really familiar with the policies but what class would the article be right now?

4490: 4105: 3210:"Examples of rings include: the set of all integers, the space of continuous functions ...". First of all, the 164:

so easily... But I will still try to get it to a decent standard. In particular, the article is lacking images.

2839: 1947:

because that is how the subject should be treated. We should make it understandable to the general population.

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the applications of ring theory to other parts of mathematics (and science). Remember, this article is about

4311:. This is just one example of how some important considerations in the subject are neglected in the lead. -- 4093: 4065: 3827: 3450: 3317: 3308:, which provide many natural examples of commutative rings, their theory, which is considered to be part of 3305: 47: 17: 2083:

Commutative rings, a ring with the property ab = ba, are much better understood than noncommutative ones.

764:. Addition and multiplication of these functions is via the addition and multiplication in each individual 5109: 5016: 4743: 4561: 4546: 4464: 4336: 4271: 4239: 4180: 3689: 3502: 3325: 3059: 2852: 2806: 2474: 2426: 2393: 2357: 2298: 2269: 2216: 2211:

Let me thank you again for letting me know why you didn't like the new version, now we can go forward. --

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I don't think that the article is supposed to cover everything (although I could be wrong). Compare it to

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Repeating "Let (R, +, ⋅) be a ring. " every time is odd. Perhaps mention it once at the top (compare with

3142:

In technical terms, a ring is an abelian group with multiplication that distributes over addition (i.e.,

1963:

Commutative rings, a ring with the property ab = ba, are much better understood than noncommutative ones

3118:

in which addition and multiplication are defined and have similar properties to those familiar from the

2494: 2016: 1868: 1832: 1780: 106:

I am a bit worried about this article. Namely, it does not contain so many of the important concepts in

2903:

Well somebody moved it down (which I like), but it's still an image of text. I've created and inserted

438:

No, everything that I am adding is from the top of my head (which I am now backing up by references).

5051: 5048: 4140: 4080: 3704: 3599: 3360: 3050:, and that examples should focus more on non-commutative cases (e.g., matrices), since we don't have 3017: 2842:) first; there is no point discussing older one, which was clearly not finished. (I'm baffled by how 2716: 2584: 2520: 2443: 1993: 1899: 1309: 1071: 1027: 958: 888: 839: 626: 402: 375: 322: 208: 147: 118: 2928:

Thanks for the image! It looks much better. But I think we need something more visual at the top. --

2913:. Feel free to play around with it (isn't it nice that you can edit it easily? (unlike the image)). 4980: 4938: 4783: 4716: 4591: 4587: 4478: 4409: 4264:

what a binary operation is: readers who do not know that much will never understand what a ring is.

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Just to start off- I don't have much of an opinion about whose version of this article is better.

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as a shortcut to understanding as a whole the cohomologies defined separately in each dimension.

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important enough to mention? (I'd never heard of the term before I came across this article.) --

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make the text easier to read. Prefer true prose where possible (this is almost always possible).

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And you think admins are the power that be? That only proves how you're new to Knowledge. --

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Another example: 1 is not the only unit in the ring of all integers: -1 is also a unit. --

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Let us now focus on Plclark's suggestions (thanks Pete, I greatly appreciate your input).

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I have now re-written (part of) the lead to address your concerns. Please have a look. --

3426:

Sigma rings are not rings in the algebraic sense. Why mention sigma rings here anyway? --

586:

Z/4 would be better represented giving multiplication and addition tables (compare with

5011:

Due to these punctuation errors, the article will be edited to correct said mistakes.--

4871: 4866:

Sorry, perhaps I wasn't clear the second paragraph was directed at the WP:OWN comment.

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ring of continuous functions vanishing at infinity of a locally compact Hausdorff space

2329:

I will add back your content now since you insist. But I strongly advise that you read

2207:

of materials on core topics), simply didn't have the rooms for such additional content.

2190: 1820: 1426:. (Except that more than one integer may correspond to a single ring element this way.) 259:

in such a way that ring operations are preserved. The formal definition is given below.

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Yeah, I'd like some more applied examples. Certainly the Lie algebra to ring thing (

1567:. Perhaps people could rewrite the text from the MacTutor archive (cited in article). 227: 4266:

I have tried to trim some of the fat, but there is a lot more left. All the best, --

4647: 4482: 4114: 3719: 3419: 2999: 1699: 1083: 1079: 4786:, the current lead seems to be of appropriate length. Opinions on the new lead? -- 2333:

and note the structure. The structure should first start with a familiar example (

4738:

should be short and simple, and other information should come in later sections.

1750:

told me at my talkpage that he wants to have the organization similar to that of

4922:

which corrects following punctuation errors in the introduction of the article:

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Like I said above, I don't see why we have to follow the format or structure of

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If you wish to start a new discussion or revive an old one, please do so on the

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OK. I understand that I must explain word by word. Here is the main criticism:

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to the article before implementing them (small changes are not a problem). --

4113:. The question is a common one, well-puzzled, and this answer is speculative. 3441:

Throughout the article, a ring is assumed to be unital and associative. (See

2884:

Why does the article begin with an image of text? This is frowned upon (see

2820:. In future, I suggest, that you discuss before you make such major edits. 2436: 1063: 664: 1596:(I have too much on my hands to do all of the above now (in real life!)). 493:, +, ⋅) will not be used for a ring since this is understood (rather 'let 3589:

P.S The other thing that you have to remember is that I am a new user...

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This paragraph does not make sense (if you ask me, it should be removed;

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really needed) should be given *after* the definitions, not before them.

2135:

applications, such as homological properties and polynomial identities.

974:, multiplication in a ring need not be commutative. Some fields such as 349:

for the dihedral group. Would it be possible to get an image like that?

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They are just a note to myself telling me what parts I should improve.

4590:

in its entirity. This link will address your remaining concerns since

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as basis. Multiplication is defined by the rules that the elements of

905:

the symbol ⋅ is usually omitted and multiplication is just denoted by

4988: 4976: 4946: 4934: 3492:(I don't respond to your argument as requested.) At least you should 183:

I have added an image of a commutative diagram to the section on the

712:

with component-wise addition and multiplication. More formally, let

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close to FA. When articles reach a good quality, it is time for FA.

2119:. Not that this is a problem but it is still not written properly. 4303:(a famous non-commutative ring theorist) and note the famous book 4104:

A ring is different than its group of units. The non-units form a

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representation ring of a finite group (mentioned above, I believe)

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I don't know if deleting some examples is a good idea. As I said,

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ring. The direct product is the collection of "infinite-tuples" (

2988:

semigroup ring k of a semigroup M (over a base field, or ring, k)

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as a tool to understand curves and surfaces. I tend to think of

1549:

The following improvements are necessary (in order of priority):

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displayed and didn't bother to check the result of their edit.

3422:. They are not considered a ring in the sense used in algebra. 1359:

as an element of the additive group of the ring (that is, 0 if

312:

in such a way that ring operations are preserved. Formally, ...

160:

Looking at it more closely, I don't think that I can get it to

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in the second ring. This is a totally unhelpful notation, as 1

3202:

A ring is one of the two main subjects of study in elementary

1509:

Above is more of the article which I think should be removed.

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of rings is simply a natural correspondence between two rings

25: 4711:

the lead, which address only the basic aspects of the theory.

4175:. Does anyone else wish to express an opinion on the matter? 847:

I didn't actually write this section (this section was there

3103:

Apparently I have to criticize your new lede (per request):

4933:(also called commutative group) under addition; and 2. a 2838:

First, about the lede. Please read the newer version (at

2439:. The "community" includes math administrators (power). 2293:

personal, but that's just, well, preference, I guess. --

5047:

soon. Unless someone has an idea what could replace it.

4163:

A sequence of revisions has recently been undertaken by

3269:) are much better understood than noncommutative ones. 4919: 4915: 4257: 4172: 4168: 3761: 3730: 575: 4132:

Out of place sentence in "Notes on Definition" section

3041:

Is this article about "basic concepts in ring theory"?

2639:. Your lede would have been much more appropriate for 1587:"Category theoretical description" should be expanded. 4171:

much clearer and more straightforward to follow than

3243: 341:

so I will try to expand the corresponding sections.

3649:

editing the article (my apologies again to Taku). --

3563:

May I politely suggest, PST, that you chill out and

639:

Thanks! I will respond to each of them once sorted.

604: 4972:(also called commutative group) under addition, and 2982:

homogeneous coordinate ring of a projective variety

2943:

New section: some examples of the ubiquity of rings

611:I think that the reference section is now perfect. 606:

contains reference information for many math books.

489:

Maybe I will write: In this article, the notation (

4088:I'll be bold and note that the etimology is vague. 3285:but you have to explain this to other readers). -- 3261: 2958:Boolean ring of clopen sets of a topological space 2756:Time and time again, I have requested you to read 2558:that you won't get the blame for those mistakes). 2421:a different format. Don't you agree with that? -- 3552: 2715:. Otherwise we might as well include stuff about 2380:Ok, then we seek inputs from other contributors. 1237:},with the following addition and multiplication: 986:. Mathematicians writing in those areas (such as 434:Be careful that you don't simply copy a textbook. 2574:my point of view (then I can understand yours). 2570:because there everything is explained clearly). 4541:And so on. Please reconsider. All the best, -- 1943:on the real numbers? I suggested that you read 1577:"Group properties of rings" has to be expanded. 1303:Note that є is a zero divisor: є ≠ 0 but є = 0. 1070:becomes a ring if we define addition to be the 3975:lack of images (does not in itself exclude GA) 2979:coordinate ring of an affine algebraic variety 2095:a and b in the ring (mathematical precision). 1188:and multiply together as they do in the group 998:to mean "commutative ring" by convention, and 2961:universal enveloping algebra of a Lie algebra 2952:tensor/symmetric/exterior algebra of a module 1656:which still omits fairly important concepts. 1433:commutes with all other elements of the ring. 8: 1528:which I will do... (please help if you can) 1406:viewed as an integer as above; second, with 574:What should be included in the article? See 5067:as the identity element of ‡, which is the 3960:, where possible, to illustrate the topic. 3328:on (non-existent) 'noncommutative spaces'. 1592:There needs to be something on Jordan rings 1418:taking place in the ring. Thus the integer 4754:it if you let me fix it first). Cheers, -- 1391:, is usually written for the ring element 90:There is another version of this article: 3445:for nonassociative cases; in particular, 3391:as I have requested you several times. -- 3242: 2970:ring completion of a commutative semiring 497:be a ring'). But this is low priority at 1422:may be identified with the ring element 321:Hope that's not discouraging..., cheers 3530: 1414:1 and multiplication in the expression 1119:, forms a ring under the operations of 3538:http://mathworld.wolfram.com/Ring.html 3355:studies the structure of rings, their 2607:I also want to advise Taku to work on 913:rules are used, so that, for example, 44:Do not edit the contents of this page. 4459:into this article?) All the best, -- 3628:confusing to everyone else, I think! 3300:Due to its intimate connections with 580:contains a few other ideas. Also see 207:times the definition is superfluous. 7: 2994:Clifford algebra of a quadratic form 1127:. This is a subring of the space of 5059:The statement in question defines 1 1894:Mainly, I would advise Taku to see 1630:Thanks for working on the article! 1375:is positive, and the opposite of (− 654:Copied text that may be added later 4918:JamesBWatson reverted the edit of 3477:that you drop your version now. -- 2643:(once corrected) because there it 2352:You didn't answer my question. -- 352:Thankyou very much for your help. 24: 4579:, if not crucial, in ring theory. 2115:This is copied (completely) from 1902:for his opinion on your changes. 1074:of sets and multiplication to be 723:. Then the direct product of the 716:be the union of all of the rings 523:Cleanup tags don't help too much. 4159:Recent recasting of this article 4012: 3992: 3978: 3968: 3944: 3928: 3918: 3894: 3884: 3863: 3849: 3839: 3812: 3798: 3410:comes from the German word 'Zahl 3334:stop copying from other articles 2955:de Rham cohomology of a manifold 1563:History has to be expanded like 1030:has been specified. Please help 1008: 1000:not necessarily commutative ring 992:Éléments de géométrie algébrique 961:has been specified. Please help 939: 891:has been specified. Please help 869: 398:How does the article look now? 29: 4013: 3993: 3979: 3969: 3945: 3929: 3919: 3895: 3885: 3864: 3850: 3846:(citations to reliable sources) 3840: 3813: 3799: 3069:I have a few points to mention: 1692:Lie ring was unfamiliar to him. 1402:coincide, that is, first, with 1201:: Let є be a formal symbol and 1049:. So does their generalization 893:improve this general definition 4773:Do I detect a slight trace of 2335:not with the formal definition 1351:an element of the ring define 583:and the subpages listed there. 1: 4876:02:43, 22 November 2009 (UTC) 4850:00:24, 22 November 2009 (UTC) 4825:18:42, 21 November 2009 (UTC) 4798:13:41, 20 November 2009 (UTC) 4766:12:38, 20 November 2009 (UTC) 4748:11:51, 20 November 2009 (UTC) 4731:01:34, 20 November 2009 (UTC) 4705:17:22, 19 November 2009 (UTC) 4153:06:16, 19 February 2009 (UTC) 4050:07:15, 19 February 2009 (UTC) 3708:11:26, 19 February 2009 (UTC) 3694:11:08, 19 February 2009 (UTC) 3659:19:58, 24 December 2008 (UTC) 3638:10:58, 28 December 2008 (UTC) 3616:16:53, 24 December 2008 (UTC) 3581:16:40, 24 December 2008 (UTC) 3522:18:07, 24 December 2008 (UTC) 3507:16:55, 24 December 2008 (UTC) 3487:16:31, 24 December 2008 (UTC) 3467:16:31, 24 December 2008 (UTC) 3436:16:31, 24 December 2008 (UTC) 3401:16:31, 24 December 2008 (UTC) 3373:universal enveloping algebras 3359:, or, in different language, 3346:16:31, 24 December 2008 (UTC) 3295:16:31, 24 December 2008 (UTC) 3229:16:31, 24 December 2008 (UTC) 3197:16:31, 24 December 2008 (UTC) 3137:16:31, 24 December 2008 (UTC) 3099:16:09, 24 December 2008 (UTC) 3064:15:28, 24 December 2008 (UTC) 3034:13:40, 24 December 2008 (UTC) 3008:13:00, 24 December 2008 (UTC) 2938:13:42, 24 December 2008 (UTC) 2923:12:56, 24 December 2008 (UTC) 2898:12:55, 23 December 2008 (UTC) 2872:16:14, 24 December 2008 (UTC) 2857:15:15, 24 December 2008 (UTC) 2833:10:31, 24 December 2008 (UTC) 2811:22:59, 23 December 2008 (UTC) 2789:10:31, 24 December 2008 (UTC) 2734:10:31, 24 December 2008 (UTC) 2713:basic concepts in ring theory 2681:10:31, 24 December 2008 (UTC) 2601:20:03, 23 December 2008 (UTC) 2537:18:24, 23 December 2008 (UTC) 2499:23:00, 22 December 2008 (UTC) 2479:22:13, 22 December 2008 (UTC) 2460:22:08, 22 December 2008 (UTC) 2431:22:02, 22 December 2008 (UTC) 2415:22:00, 22 December 2008 (UTC) 2398:21:25, 22 December 2008 (UTC) 2376:21:18, 22 December 2008 (UTC) 2362:21:07, 22 December 2008 (UTC) 2347:20:59, 22 December 2008 (UTC) 2325:20:59, 22 December 2008 (UTC) 2303:20:54, 22 December 2008 (UTC) 2288:20:51, 22 December 2008 (UTC) 2274:20:49, 22 December 2008 (UTC) 2259:20:51, 22 December 2008 (UTC) 2241:20:08, 22 December 2008 (UTC) 2221:19:55, 22 December 2008 (UTC) 2181:20:06, 22 December 2008 (UTC) 2168:20:06, 22 December 2008 (UTC) 2152:20:06, 22 December 2008 (UTC) 2129:20:06, 22 December 2008 (UTC) 2105:20:06, 22 December 2008 (UTC) 2078:20:06, 22 December 2008 (UTC) 2059:20:06, 22 December 2008 (UTC) 2021:19:50, 22 December 2008 (UTC) 1998:19:37, 22 December 2008 (UTC) 1981:19:33, 22 December 2008 (UTC) 1931:18:23, 22 December 2008 (UTC) 1912:18:02, 22 December 2008 (UTC) 1889:17:49, 22 December 2008 (UTC) 1873:17:43, 22 December 2008 (UTC) 1853:20:00, 22 December 2008 (UTC) 1837:19:01, 22 December 2008 (UTC) 1801:18:45, 22 December 2008 (UTC) 1785:17:43, 22 December 2008 (UTC) 1764:17:37, 22 December 2008 (UTC) 1742:17:25, 22 December 2008 (UTC) 1724:15:21, 22 December 2008 (UTC) 1708:14:26, 22 December 2008 (UTC) 1679:13:54, 22 December 2008 (UTC) 1649:however are fairly common). 1625:13:41, 22 December 2008 (UTC) 1606:11:14, 22 December 2008 (UTC) 1540:17:19, 21 December 2008 (UTC) 1524:Needs motivation for concepts 1519:10:48, 22 December 2008 (UTC) 1338:10:42, 22 December 2008 (UTC) 1184:commute with the elements of 982:are primarily concerned with 861:21:16, 19 December 2008 (UTC) 843:20:28, 19 December 2008 (UTC) 649:20:48, 19 December 2008 (UTC) 635:21:34, 18 December 2008 (UTC) 621:20:48, 19 December 2008 (UTC) 564:20:52, 19 December 2008 (UTC) 538:20:52, 19 December 2008 (UTC) 474:20:52, 19 December 2008 (UTC) 448:20:48, 19 December 2008 (UTC) 419:20:13, 18 December 2008 (UTC) 392:14:56, 18 December 2008 (UTC) 364:14:53, 18 December 2008 (UTC) 331:11:11, 18 December 2008 (UTC) 230:. So, for example, instead of 222:this, it is critical to stay 217:11:02, 18 December 2008 (UTC) 197:10:51, 18 December 2008 (UTC) 179:10:40, 18 December 2008 (UTC) 156:10:31, 18 December 2008 (UTC) 135:17:30, 17 December 2008 (UTC) 92:Talk:Ring (mathematics)/Draft 4952:3. such that multiplication 4910:Edit on 12:29, 31 March 2010 4905:03:38, 8 December 2009 (UTC) 4123:00:56, 1 February 2009 (UTC) 4098:22:16, 29 January 2009 (UTC) 4084:02:20, 28 January 2009 (UTC) 4070:02:09, 28 January 2009 (UTC) 4034:17:07, 26 January 2009 (UTC) 3680:17:05, 18 January 2009 (UTC) 3453:for additional terminology. 3363:, special classes of rings ( 2886:Knowledge:MOS#Images_as_text 2544:Copied from Taku's talk page 1808:universal enveloping algebra 1582:The lede should be expanded. 5026:Wrong statement in article? 4252:Article is not encyclopedic 2973:K_0 of a commutative ring R 2562:possible (don't postpone). 1728:Ok, I put a new version at 1387:is negative.) The integer, 603:templates or similar ones. 298:is a map between two rings 228:Knowledge is not a textbook 5132: 5021:18:51, 31 March 2010 (UTC) 4671:15:08, 5 August 2009 (UTC) 4656:23:06, 4 August 2009 (UTC) 4622:06:51, 5 August 2009 (UTC) 4606:06:49, 5 August 2009 (UTC) 4586:Lastly, I urge you to see 4566:01:08, 5 August 2009 (UTC) 4551:00:54, 5 August 2009 (UTC) 4517:00:33, 5 August 2009 (UTC) 4469:22:39, 4 August 2009 (UTC) 4446:10:58, 4 August 2009 (UTC) 4429:10:52, 4 August 2009 (UTC) 4395:11:02, 4 August 2009 (UTC) 4375:10:18, 4 August 2009 (UTC) 4358:09:59, 4 August 2009 (UTC) 4341:07:36, 4 August 2009 (UTC) 4323:05:06, 4 August 2009 (UTC) 4294:05:02, 4 August 2009 (UTC) 4276:02:24, 4 August 2009 (UTC) 3553:A lurker's perspective on 3320:and with the discovery of 3273:It is not understood that 2976:K_0 of a topological space 2091:it should be 'ab = ba for 1775:discussion was one-sided. 994:) frequently use the word 4994:such that multiplication 4681:the following two facts: 3237:(rings with the property 1748:User:Point-set topologist 1452:are ring elements, then ( 1410:meaning the ring element 670:and collection of rings ( 5114:10:07, 26 May 2010 (UTC) 5055:08:34, 26 May 2010 (UTC) 5041:08:14, 26 May 2010 (UTC) 4949:) under multiplication; 4491:Jacobson density theorem 4473:I might be mistaken but 4106:mathematical singularity 3547:Copied from my talk page 2840:Ring_(mathematics)/Draft 1730:Ring (mathematics)/Draft 1487:is a ring element, then 1355:as one would by viewing 1078:. This corresponds to a 876:This general definition 663:More generally, for any 4991:) under multiplication, 4244:12:55, 6 May 2009 (UTC) 4225:23:50, 5 May 2009 (UTC) 4208:11:33, 5 May 2009 (UTC) 4185:11:17, 5 May 2009 (UTC) 3925:(all significant views) 3788:reasonably well written 3451:glossary of ring theory 3318:noncommutative geometry 3306:algebraic number theory 1398:The two definitions of 1082:and is an example of a 1045:form a ring, as do the 921:is an abbreviation for 750:with the property that 738:is the set of all maps 18:Talk:Ring (mathematics) 5001: 4958: 3985:(non-free images have 3965:(tagged and captioned) 3377:homological properties 3263: 2985:Chow ring of a variety 2647:summarize the article. 2580:Point-set topologist 1545:Improvements necessary 1103:forms a ring (even an 988:Alexander Grothendieck 453:Focus on interesting, 4962: 4924: 4676:Simplify Introduction 4305:Non-commutative rings 3915:(fair representation) 3906:neutral point of view 3874:broad in its coverage 3381:polynomial identities 3264: 3262:{\displaystyle ab=ba} 2566:suggested you to see 2513:Point-set topologist 2435:edit conflict: ::see 1310:split-complex numbers 909:. Also, the standard 42:of past discussions. 4920:07:12, 31 March 2010 4916:12:29, 31 March 2010 4169:the original version 4165:Point-set_topologist 3651:Point-set topologist 3604:Point-set topologist 3514:Point-set topologist 3479:Point-set topologist 3459:Point-set topologist 3428:Point-set topologist 3393:Point-set topologist 3338:Point-set topologist 3287:Point-set topologist 3241: 3221:Point-set topologist 3189:Point-set topologist 3129:Point-set topologist 3091:Point-set topologist 3022:Point-set topologist 2967:Witt ring of a field 2930:Point-set topologist 2908:Algebraic structures 2864:Point-set topologist 2825:Point-set topologist 2781:Point-set topologist 2726:Point-set topologist 2673:Point-set topologist 2589:Point-set topologist 2525:Point-set topologist 2448:Point-set topologist 2407:Point-set topologist 2368:Point-set topologist 2339:Point-set topologist 2317:Point-set topologist 2280:Point-set topologist 2251:Point-set topologist 2233:Point-set topologist 2173:Point-set topologist 2160:Point-set topologist 2144:Point-set topologist 2121:Point-set topologist 2097:Point-set topologist 2070:Point-set topologist 2051:Point-set topologist 1973:Point-set topologist 1904:Point-set topologist 1900:User:Jakob.scholbach 1671:Point-set topologist 1598:Point-set topologist 1532:Point-set topologist 1511:Point-set topologist 1330:Point-set topologist 1207:ring of dual numbers 1072:symmetric difference 1032:improve this article 1021:to meet Knowledge's 963:improve this article 952:to meet Knowledge's 882:to meet Knowledge's 853:Point-set topologist 641:Point-set topologist 613:Point-set topologist 556:Point-set topologist 530:Point-set topologist 466:Point-set topologist 440:Point-set topologist 407:Point-set topologist 380:Point-set topologist 356:Point-set topologist 189:Point-set topologist 171:Point-set topologist 123:Point-set topologist 94:. (23 December 2008) 4784:group (mathematics) 4717:group (mathematics) 4592:group (mathematics) 4588:group (mathematics) 4479:Group (mathematics) 4410:group (mathematics) 4256:The article in its 3987:fair use rationales 3443:nonassociative ring 3310:commutative algebra 3212:set of all integers 3116:algebraic structure 3087:group (mathematics) 3079:group (mathematics) 3052:noncommutative ring 2762:Group (mathematics) 2758:group (mathematics) 2666:. In fact, this is 2568:group (mathematics) 2403:Group (mathematics) 2331:group (mathematics) 2039:Group (mathematics) 1945:group (mathematics) 1919:group (mathematics) 1896:group (mathematics) 1812:representation ring 1752:group (mathematics) 1661:group (mathematics) 1654:group (mathematics) 1565:group (mathematics) 1556:group (mathematics) 1347:is an integer, and 1121:point-wise addition 1105:associative algebra 1062:is a set, then the 1047:Eisenstein integers 976:commutative algebra 588:field (mathematics) 347:Group (mathematics) 338:Group (mathematics) 162:Group (mathematics) 112:group (mathematics) 3823:factually accurate 3701:Field of fractions 3699:You could call it 3555:ring (mathematics) 3475:strongly recommend 3302:algebraic geometry 3259: 2818:without discussion 2552:ring (mathematics) 1483:is an integer and 980:algebraic geometry 911:order of operation 659:General definition 499:this point in time 4782:In comparison to 4173:the final version 4143:comment added by 4038:Please review. -- 3664:Concepts required 3619: 3602:comment added by 3565:assume good faith 3235:Commutative rings 3037: 3020:comment added by 2604: 2587:comment added by 2540: 2523:comment added by 2463: 2446:comment added by 2036: 1686:User:TakuyaMurata 1646:algebraic variety 1444:are integers and 1429:The ring element 1363:is 0, the sum of 1043:Gaussian integers 1038: 1037: 1023:quality standards 1002:to mean "ring". 984:commutative rings 969: 968: 954:quality standards 899: 898: 884:quality standards 422: 405:comment added by 395: 378:comment added by 296:ring homomorphism 277:Formal definition 138: 121:comment added by 82: 81: 54: 53: 48:current talk page 5123: 4902: 4897: 4847: 4842: 4795: 4790: 4763: 4758: 4728: 4723: 4619: 4614: 4603: 4598: 4514: 4509: 4495:Jacobson radical 4443: 4438: 4426: 4421: 4392: 4387: 4372: 4367: 4355: 4350: 4320: 4315: 4291: 4286: 4222: 4217: 4205: 4200: 4155: 4047: 4042: 4031: 4026: 4016: 4015: 3996: 3995: 3982: 3981: 3972: 3971: 3948: 3947: 3932: 3931: 3922: 3921: 3898: 3897: 3888: 3887: 3867: 3866: 3853: 3852: 3843: 3842: 3816: 3815: 3802: 3801: 3766: 3757: 3738: 3677: 3672: 3618: 3596: 3540: 3535: 3449:.) See also the 3268: 3266: 3265: 3260: 3217:topological ring 3204:abstract algebra 3048:commutative ring 3036: 3014: 2912: 2906: 2671:corrections). -- 2603: 2581: 2539: 2517: 2462: 2440: 2030: 1825:cohomology rings 1213:, is defined as 1113:topological ring 1012: 1011: 1005: 943: 942: 936: 873: 872: 866: 602: 596: 482:commutative ring 421: 399: 394: 372: 144:commutative ring 137: 115: 68: 56: 55: 33: 32: 26: 5131: 5130: 5126: 5125: 5124: 5122: 5121: 5120: 5103: 5102: 5095: 5094: 5078: 5077: 5066: 5065: 5028: 4912: 4900: 4895: 4890: 4845: 4840: 4793: 4788: 4761: 4756: 4726: 4721: 4678: 4663: 4640: 4617: 4612: 4601: 4596: 4512: 4507: 4441: 4436: 4424: 4419: 4390: 4385: 4370: 4365: 4353: 4348: 4318: 4313: 4289: 4284: 4254: 4220: 4215: 4203: 4198: 4191:I had done (in 4161: 4138: 4134: 4090:—Ben FrantzDale 4062:—Ben FrantzDale 4057: 4045: 4040: 4029: 4024: 3903:It follows the 3881:(major aspects) 3747: 3724: 3718: 3716: 3675: 3670: 3666: 3646: 3644:Recent disputes 3597: 3558: 3544: 3543: 3536: 3532: 3357:representations 3239: 3238: 3043: 3015: 2945: 2910: 2904: 2882: 2613:not major edits 2582: 2518: 2486: 2441: 1860: 1547: 1526: 1156:is a ring, the 1131:functions from 1099:defined on the 1091:The set of all 1009: 940: 870: 833: 825: 816: 806: 797: 788: 769: 762: 755: 728: 721: 711: 700: 688: 678: 656: 627:Jakob.scholbach 600: 594: 428: 426:Informal review 400: 373: 323:Jakob.scholbach 311: 304: 258: 251: 209:Jakob.scholbach 148:Jakob.scholbach 116: 104: 87: 64: 30: 22: 21: 20: 12: 11: 5: 5129: 5127: 5119: 5118: 5117: 5116: 5098: 5097: 5090: 5089: 5081:multiplicative 5073: 5072: 5061: 5060: 5027: 5024: 5000: 4999: 4998:over addition. 4992: 4987:property is a 4973: 4956:over addition. 4945:property is a 4911: 4908: 4889: 4888:Recent changes 4886: 4885: 4884: 4883: 4882: 4881: 4880: 4879: 4878: 4857: 4856: 4855: 4854: 4853: 4852: 4830: 4829: 4828: 4827: 4810: 4809: 4808: 4807: 4801: 4800: 4780: 4779: 4778: 4751: 4750: 4734: 4733: 4712: 4692: 4691: 4687: 4686: 4677: 4674: 4662: 4659: 4639: 4636: 4635: 4634: 4633: 4632: 4631: 4630: 4629: 4628: 4627: 4626: 4625: 4624: 4584: 4580: 4577:very important 4572: 4540: 4537: 4534: 4531: 4522: 4521: 4520: 4519: 4498: 4486: 4471: 4454: 4451: 4406: 4405: 4404: 4403: 4402: 4401: 4400: 4399: 4398: 4397: 4381: 4329: 4309:I. N. Herstein 4296: 4265: 4262: 4253: 4250: 4249: 4248: 4247: 4246: 4228: 4227: 4210: 4160: 4157: 4133: 4130: 4128: 4126: 4125: 4102: 4101: 4100: 4056: 4053: 4020: 4019: 4018: 4017: 3999: 3998: 3997: 3951: 3950: 3949: 3935: 3934: 3933: 3901: 3900: 3899: 3870: 3869: 3868: 3819: 3818: 3817: 3783: 3782: 3767: 3715: 3712: 3711: 3710: 3665: 3662: 3645: 3642: 3641: 3640: 3557: 3551: 3542: 3541: 3529: 3528: 3527: 3526: 3525: 3524: 3470: 3469: 3439: 3438: 3416:measure theory 3404: 3403: 3369:division rings 3349: 3348: 3322:quantum groups 3298: 3297: 3258: 3255: 3252: 3249: 3246: 3232: 3231: 3200: 3199: 3140: 3139: 3071: 3070: 3042: 3039: 2998:And so forth. 2996: 2995: 2992: 2989: 2986: 2983: 2980: 2977: 2974: 2971: 2968: 2965: 2962: 2959: 2956: 2953: 2944: 2941: 2926: 2925: 2881: 2880:Image of text? 2878: 2877: 2876: 2875: 2874: 2794: 2793: 2792: 2791: 2769: 2768: 2767: 2766: 2751: 2750: 2745: 2744: 2739: 2738: 2737: 2736: 2706: 2705: 2704: 2703: 2696: 2695: 2686: 2685: 2684: 2683: 2651: 2650: 2649: 2648: 2625: 2624: 2485: 2482: 2467: 2466: 2465: 2464: 2350: 2349: 2327: 2309: 2308: 2307: 2306: 2305: 2261: 2245: 2209: 2208: 2201: 2198: 2155: 2154: 2132: 2131: 2108: 2107: 2081: 2080: 2062: 2061: 2028: 2027: 2026: 2025: 2024: 2023: 2003: 2002: 2001: 2000: 1961: 1960: 1955: 1954: 1949: 1948: 1939: 1938: 1937: 1936: 1892: 1891: 1859: 1856: 1840: 1839: 1821:function field 1788: 1787: 1772: 1767: 1766: 1711: 1710: 1694: 1693: 1628: 1627: 1594: 1593: 1589: 1588: 1584: 1583: 1579: 1578: 1574: 1573: 1569: 1568: 1560: 1559: 1546: 1543: 1525: 1522: 1507: 1506: 1505: 1504: 1477: 1434: 1427: 1326: 1325: 1305: 1304: 1302: 1272: 1238: 1194: 1193: 1141: 1140: 1125:multiplication 1088: 1087: 1055: 1054: 1036: 1035: 1028:cleanup reason 1013: 972:As noted below 967: 966: 959:cleanup reason 944: 897: 896: 889:cleanup reason 874: 864: 863: 845: 835: 829: 821: 812: 802: 793: 784: 767: 760: 753: 726: 719: 703: 698: 692:direct product 680: 674: 655: 652: 624: 623: 608: 607: 591: 584: 572: 567: 566: 551: 550: 541: 540: 525: 524: 521: 520: 519: 516: 513: 510: 506:Markup-style: 503: 502: 486: 485: 477: 476: 460: 459: 436: 435: 427: 424: 334: 333: 319: 318: 317: 316: 315: 314: 313: 309: 302: 287:I would write 285: 284: 283: 282: 281: 280: 279: 267: 266: 265: 264: 263: 262: 261: 260: 256: 249: 232: 231: 219: 168: 167: 166: 165: 103: 100: 98: 96: 95: 86: 83: 80: 79: 74: 69: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 5128: 5115: 5111: 5107: 5101: 5093: 5086: 5082: 5076: 5070: 5064: 5058: 5057: 5056: 5053: 5050: 5045: 5044: 5043: 5042: 5038: 5034: 5025: 5023: 5022: 5018: 5014: 5009: 5005: 4997: 4993: 4990: 4986: 4985:invertibility 4982: 4978: 4974: 4971: 4970:abelian group 4967: 4966: 4965: 4961: 4957: 4955: 4950: 4948: 4944: 4943:invertibility 4940: 4936: 4932: 4931:abelian group 4927: 4923: 4921: 4917: 4909: 4907: 4906: 4903: 4898: 4887: 4877: 4873: 4869: 4865: 4864: 4863: 4862: 4861: 4860: 4859: 4858: 4851: 4848: 4843: 4836: 4835: 4834: 4833: 4832: 4831: 4826: 4822: 4818: 4814: 4813: 4812: 4811: 4805: 4804: 4803: 4802: 4799: 4796: 4791: 4785: 4781: 4776: 4772: 4771: 4770: 4769: 4768: 4767: 4764: 4759: 4749: 4745: 4741: 4736: 4735: 4732: 4729: 4724: 4718: 4713: 4709: 4708: 4707: 4706: 4702: 4698: 4689: 4688: 4684: 4683: 4682: 4675: 4673: 4672: 4669: 4660: 4658: 4657: 4653: 4649: 4645: 4637: 4623: 4620: 4615: 4609: 4608: 4607: 4604: 4599: 4593: 4589: 4585: 4581: 4578: 4573: 4569: 4568: 4567: 4563: 4559: 4554: 4553: 4552: 4548: 4544: 4528: 4527: 4526: 4525: 4524: 4523: 4518: 4515: 4510: 4504: 4499: 4496: 4492: 4487: 4484: 4480: 4476: 4472: 4470: 4466: 4462: 4458: 4449: 4448: 4447: 4444: 4439: 4433: 4432: 4431: 4430: 4427: 4422: 4416: 4411: 4396: 4393: 4388: 4382: 4378: 4377: 4376: 4373: 4368: 4361: 4360: 4359: 4356: 4351: 4344: 4343: 4342: 4338: 4334: 4326: 4325: 4324: 4321: 4316: 4310: 4306: 4302: 4297: 4295: 4292: 4287: 4280: 4279: 4278: 4277: 4273: 4269: 4259: 4258:present state 4251: 4245: 4241: 4237: 4232: 4231: 4230: 4229: 4226: 4223: 4218: 4211: 4209: 4206: 4201: 4194: 4189: 4188: 4187: 4186: 4182: 4178: 4174: 4170: 4166: 4158: 4156: 4154: 4150: 4146: 4145:4.240.213.232 4142: 4131: 4129: 4124: 4120: 4116: 4112: 4107: 4103: 4099: 4095: 4091: 4087: 4086: 4085: 4082: 4078: 4074: 4073: 4072: 4071: 4067: 4063: 4054: 4052: 4051: 4048: 4043: 4036: 4035: 4032: 4027: 4010: 4006: 4005: 4003: 4000: 3990: 3988: 3976: 3966: 3962: 3961: 3959: 3958: 3952: 3943: 3942: 3940: 3936: 3926: 3916: 3912: 3911: 3909: 3907: 3902: 3892: 3882: 3878: 3877: 3875: 3871: 3861: 3859: 3847: 3837: 3833: 3832: 3830: 3829: 3824: 3820: 3810: 3808: 3796: 3792: 3791: 3789: 3785: 3784: 3781:for criteria) 3780: 3776: 3774: 3770: 3769: 3768: 3765: 3764: 3760: 3755: 3751: 3746: 3745: 3741: 3736: 3732: 3728: 3723: 3722: 3713: 3709: 3706: 3702: 3698: 3697: 3696: 3695: 3691: 3687: 3682: 3681: 3678: 3673: 3663: 3661: 3660: 3656: 3652: 3643: 3639: 3635: 3631: 3627: 3622: 3621: 3620: 3617: 3613: 3609: 3605: 3601: 3593: 3590: 3587: 3583: 3582: 3578: 3574: 3570: 3566: 3561: 3556: 3550: 3548: 3539: 3534: 3531: 3523: 3519: 3515: 3510: 3509: 3508: 3504: 3500: 3495: 3491: 3490: 3489: 3488: 3484: 3480: 3476: 3468: 3464: 3460: 3456: 3455: 3454: 3452: 3448: 3444: 3437: 3433: 3429: 3425: 3424: 3423: 3421: 3420:rings of sets 3417: 3413: 3409: 3402: 3398: 3394: 3390: 3386: 3385: 3384: 3382: 3378: 3374: 3370: 3366: 3362: 3358: 3354: 3347: 3343: 3339: 3335: 3331: 3330: 3329: 3327: 3323: 3319: 3315: 3311: 3307: 3303: 3296: 3292: 3288: 3284: 3280: 3276: 3272: 3271: 3270: 3256: 3253: 3250: 3247: 3244: 3236: 3230: 3226: 3222: 3218: 3213: 3209: 3208: 3207: 3205: 3198: 3194: 3190: 3185: 3184: 3183: 3181: 3177: 3173: 3169: 3165: 3161: 3157: 3153: 3149: 3145: 3138: 3134: 3130: 3125: 3124: 3123: 3121: 3117: 3113: 3109: 3104: 3101: 3100: 3096: 3092: 3088: 3082: 3080: 3075: 3068: 3067: 3066: 3065: 3061: 3057: 3053: 3049: 3040: 3038: 3035: 3031: 3027: 3023: 3019: 3010: 3009: 3005: 3001: 2993: 2990: 2987: 2984: 2981: 2978: 2975: 2972: 2969: 2966: 2963: 2960: 2957: 2954: 2951: 2950: 2949: 2942: 2940: 2939: 2935: 2931: 2924: 2920: 2916: 2909: 2902: 2901: 2900: 2899: 2895: 2891: 2887: 2879: 2873: 2869: 2865: 2860: 2859: 2858: 2854: 2850: 2845: 2841: 2837: 2836: 2835: 2834: 2830: 2826: 2821: 2819: 2813: 2812: 2808: 2804: 2799: 2790: 2786: 2782: 2778: 2773: 2772: 2771: 2770: 2763: 2759: 2755: 2754: 2753: 2752: 2747: 2746: 2741: 2740: 2735: 2731: 2727: 2722: 2718: 2714: 2710: 2709: 2708: 2707: 2700: 2699: 2698: 2697: 2693: 2688: 2687: 2682: 2678: 2674: 2669: 2665: 2660: 2655: 2654: 2653: 2652: 2646: 2642: 2638: 2634: 2629: 2628: 2627: 2626: 2621: 2620: 2619: 2616: 2614: 2610: 2605: 2602: 2598: 2594: 2590: 2586: 2578: 2575: 2571: 2569: 2563: 2559: 2555: 2553: 2547: 2545: 2541: 2538: 2534: 2530: 2526: 2522: 2514: 2511: 2508: 2504: 2501: 2500: 2496: 2492: 2483: 2481: 2480: 2476: 2472: 2461: 2457: 2453: 2449: 2445: 2438: 2434: 2433: 2432: 2428: 2424: 2419: 2418: 2417: 2416: 2412: 2408: 2404: 2400: 2399: 2395: 2391: 2386: 2381: 2378: 2377: 2373: 2369: 2364: 2363: 2359: 2355: 2348: 2344: 2340: 2336: 2332: 2328: 2326: 2322: 2318: 2314: 2310: 2304: 2300: 2296: 2291: 2290: 2289: 2285: 2281: 2277: 2276: 2275: 2271: 2267: 2262: 2260: 2256: 2252: 2248: 2247: 2246: 2243: 2242: 2238: 2234: 2230: 2227: 2223: 2222: 2218: 2214: 2206: 2202: 2199: 2196: 2192: 2188: 2187: 2186: 2183: 2182: 2178: 2174: 2170: 2169: 2165: 2161: 2153: 2149: 2145: 2141: 2140: 2139: 2136: 2130: 2126: 2122: 2118: 2114: 2113: 2112: 2106: 2102: 2098: 2094: 2090: 2086: 2085: 2084: 2079: 2075: 2071: 2067: 2066: 2065: 2060: 2056: 2052: 2047: 2046: 2045: 2042: 2040: 2034: 2033:edit conflict 2022: 2018: 2014: 2009: 2008: 2007: 2006: 2005: 2004: 1999: 1995: 1991: 1987: 1986: 1985: 1984: 1983: 1982: 1978: 1974: 1970: 1967: 1964: 1957: 1956: 1951: 1950: 1946: 1941: 1940: 1934: 1933: 1932: 1928: 1924: 1920: 1916: 1915: 1914: 1913: 1909: 1905: 1901: 1897: 1890: 1886: 1882: 1877: 1876: 1875: 1874: 1870: 1866: 1857: 1855: 1854: 1850: 1846: 1838: 1834: 1830: 1826: 1822: 1817: 1816:Burnside ring 1813: 1809: 1805: 1804: 1803: 1802: 1798: 1794: 1786: 1782: 1778: 1773: 1769: 1768: 1765: 1761: 1757: 1753: 1749: 1746: 1745: 1744: 1743: 1739: 1735: 1731: 1726: 1725: 1721: 1717: 1709: 1705: 1701: 1696: 1695: 1691: 1687: 1683: 1682: 1681: 1680: 1676: 1672: 1668: 1666: 1662: 1657: 1655: 1650: 1647: 1643: 1639: 1635: 1631: 1626: 1622: 1618: 1614: 1610: 1609: 1608: 1607: 1603: 1599: 1591: 1590: 1586: 1585: 1581: 1580: 1576: 1575: 1571: 1570: 1566: 1562: 1561: 1557: 1552: 1551: 1550: 1544: 1542: 1541: 1537: 1533: 1529: 1523: 1521: 1520: 1516: 1512: 1502: 1498: 1494: 1490: 1486: 1482: 1478: 1475: 1471: 1467: 1463: 1459: 1455: 1451: 1447: 1443: 1439: 1435: 1432: 1428: 1425: 1421: 1417: 1413: 1409: 1405: 1401: 1397: 1396: 1394: 1390: 1386: 1382: 1378: 1374: 1370: 1366: 1362: 1358: 1354: 1350: 1346: 1342: 1341: 1340: 1339: 1335: 1331: 1323: 1319: 1315: 1311: 1307: 1306: 1300: 1296: 1292: 1288: 1284: 1280: 1276: 1270: 1266: 1262: 1258: 1254: 1250: 1246: 1242: 1236: 1232: 1228: 1224: 1220: 1216: 1212: 1208: 1205:a field. 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3731:visual edit 3598:—Preceding 3494:acknowledge 3389:ring theory 3365:group rings 3353:ring theory 3108:mathematics 3016:—Preceding 2664:ring theory 2641:ring theory 2637:ring theory 2583:—Preceding 2519:—Preceding 2491:JackSchmidt 2442:—Preceding 2195:Ring theory 2117:ring theory 2013:JackSchmidt 1865:JackSchmidt 1858:Big changes 1829:JackSchmidt 1777:JackSchmidt 1170:free module 1051:Kummer ring 1034:if you can. 965:if you can. 895:if you can. 401:—Preceding 374:—Preceding 117:—Preceding 108:ring theory 36:This is an 4838:lately. -- 4301:P. M. Cohn 4081:Algebraist 3828:verifiable 3705:Algebraist 1367:copies of 1158:group ring 1093:continuous 840:Algebraist 185:definition 4661:Etymology 4380:version)? 4075:This was 4009:Pass/Fail 3955:contains 3891:(focused) 3714:GA Review 3406:The word 3326:functions 2777:talk page 2503:Hi Jack, 2437:democracy 1634:Lie rings 1395:1. 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