706:'s assessment is correct. The table is a bit of an off-topic intruder in the body of the article, especially as the first paragraph. This is already a good justification to the rollback. Secondarily (just my opinion, that is) it is not a very good table. It seems to suggest that the two notations of addition and multiplication have something to do with the definition of these structures, but that isn't true. It's also half-empty, for whatever reason. Frankly, nothing like this appears in any mathematics textbook. It makes more sense to have structures split apart by number of operations (and I believe that others have successfully contributed such charts.) I do recognize what it is, however. I've drawn many such pictures while learning my way around new disciplines. And I understand it's nice to share such things, but quite often they are not suitable for a wiki math article: they tend to be shaped/organized according to the note-taker's personal ideas.
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664:. For including it there, I suggests to remove the heading "Examples" as well as the headings of its subsections, and to make sections from the bold texts such as "simple structures", "group-like structures, "ring-like structures", ... This would allows adding some text explaining that this is a classification by the number of operations and their arity, and also adding tables, like yours for clarifying the relations between structures. When this would be done, one could also modify the template
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529:- however, there, the table risks becoming bloated by a complete taxonomy of all algebraic structures. That may be useful in itself, but rings and groups are very important and basic concepts that occur again and again, and they have an associated set of terminology (which is rather confusing, particularly to nonmathematicians who are unused to memorising large amounts of terminology).
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two binary operators (typically called "addition" and "multiplication"), satisfying the following: ... ." I think the article might be easier to read and understand if this kind of definition appeared early in the article. Someone with a ring theory book handy should be able to provide this with a reference. Unfortunately, I don't have such handy. Thanks,
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It's inclusion, but it doesn't make much sense to use subset symbol, because there aren't any standard "symbols" for "the class of all
Euclidean domains", e.g., unless you want to make up several just for this article. Writing "Euclidean domain contained in PID contained in UFD, etc." is misleading,
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What is the definition (or what are the axioms) of a ring? Are they clearly stated in this article? I couldn't find them in a quick review. If they are there, please excuse me for asking. However, I think the first section after the intro / lede should say something like, "A ring is a set with
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A ring is called commutative if its multiplication is commutative. The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in
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does not contain the same information. It is just a list of various structures and does not state the critical properties which distinguish them. Also, as I have explained, there is good reason to focus this information only on the structures culminating in fields. The template has too many
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I was wrong, the ring article doesn't require it, it requires unity. My fault. (Although I disagree with unity requirement, as well, that's another matter.) Actually, I just disagree with these universal wikipedia definitions (rather than article by article basis).
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I added a table near the top of this article, giving a summary of the principal ring-like and group-like structures, and the distinctions between them. I think this table is a very valuable summary which should appear somewhere prominent on
Knowledge.
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Sorry for exaggerating in my edit comment -- the patent nonsense was only in
Knowledge for a little over a day before I reverted it. (I misread the date.) I find it embarrassing, though, that someone who trusts Knowledge asked me for an explanation of
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over any of these previous examples are also rings with matrix addition and multiplication. Try to chase some of the links on the ring theory page to arrive at more specific pages: they are likely to have other concrete examples.
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There are important examples of rings that do not have a identity. Regardless, and identity can always be formally adjoined by using the adjoint of the forgetful functor from the category of rings with unity to the category of
749:, an article linked to in the first sentence of this article. This article contains many examples that are not always called "examples", especially when they are sufficiently important to be the subject of a specific section.
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It seems to me it might more usefully appear either later in a number of pages in this field or as a page of its own, prominently wikilinked from here and others. As a side note, it's also a bit afflicted with Caps
Disease.
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article, as a ring is the most "advanced" mathematical concept in this table. I think this is an appropriate place to draw the line, because the various other specialisations of rings are, well, more specialised.
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too, at least, and maybe in some of the specific articles. It's a big bulky to make into an infobox and I don't see how to shrink it without losing its usefulness. As for "caps disease", feel free to
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There is inconsistency with the definition of "Ring" in the main article about rings. As far as I know a ring is assumed to have an identity unless stated otherwise and not the other way around.
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I don't know much about ring theory, other than that it keeps popping up on wikipedia everywhere... Could someone give some examples of rings and the 2 binary operators?
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Except that the ring article contradicts this and requires commutativity, which adds credence to my belief that this shouldn't be required in the definition.
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I am going to revert D. Lazard's revert. I would like to ask D.Lazard to please discuss the matter here instead of simply reverting my change again.
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Here is my site with ring theory example problems. Someone please put this link in the external links section if you think it's helpful and relevant.
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The first two sentences are awful. Repeating the idea that ring theory is the study of rings. Very informative - not. Too much premature jargon.
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Where do you think it should go specifically? Should we make a template out of it? I think it could usefully appear in the article
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the word "example" appears no more than three times in this article, yet examples of rings are what makes ring theory important.
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reverted my contribution, simply saying "This does not belongs to this article". I think this was contrary to
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because it makes it sound like a ED is set-theoretically contained in a PID, contained in a UFD, not the same.
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No, multiplication in rings is not generally required to be commutative. I double-checked this in
Herstein's
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for linking phrases such as "ring-like structures" to the corresponding section of
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Talk:Ring_(mathematics)#Sorting_out_ring_theory_and_ring_.28mathematics.29
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Is the above a hierarchy of inclusion? If so, use the subset symbol. --
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Talk:Ring (mathematics)/Archive 4#merging ring theory into this article
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http://www.exampleproblems.com/index.php/Abstract_Algebra#Rings
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all under their ordinary addition and multiplication. Square
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Again, this content is not about ring theory, but about
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