Talk:Module (mathematics)

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very specific set of axioms is important for studying a new kind of interesting scenario. Dropping all axioms ultimately leads you to be able to prove absolutely nothing. Just because something is more general, does not mean it's more "powerful", in fact it's quite the opposite! The more general a theory is, the weaker it becomes. But on the flipside you can apply it to more and more different scenarios. So that's what's going on. Abelian groups are WEAKER but MORE GENERAL than modules, however the generality is very superficial because you could define some trivial scalar multiplication (such as the one with integers) and suddenly any results about modules (which could potentially be stronger statements) also apply to your new structure that used to be an abelian group.

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and any function you choose that satisfies some criteria that the previous function satisfied). In fact, I think this is an excellent idea, since in practice you often do need the scalar multiplication when working with abelian groups, and that's why I've seen many authors define it as a "notational shorthand". Would it make more sense to include it in the definition? In my opinion, absolutely! However, that's unfortunately not the current world we live in, and at present an abelian group is a very barebones set of axioms that, as far as I can see, is just an R-module with some of the axioms dropped out (and thus a generalization).

792:(2) The citation to the Knowledge Arbitration Committee is about style. It's difficult to believe that an author's style is to surrender control for more rendering variability, where the author is less aware of whether their formulas will even appear correct. Will the identity matrix look more like the number 1 depending on browser interactions? Is that a question of style? And is the homogeneity of encoding a matter of screen reader accessibility (heterogeneous markup is a death sentence to screen readers)? What is this concept of style to a screen reader? 293: 283: 262: 229: 372: 789:, which suggests that while large-scale changes may be controversial, one should refer to clear improvements: "Proposed changes should generally be discussed on the talk page of the article before implementation. If there is no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as WikiProject Mathematics for mathematical articles." 220: 1659:

Bob's world, however a statement in Bob's world does not necessarily carry over to Alice's world. What if Bob makes a statement about the ring R? Alice's world doesn't even have the language to talk about two different operations between elements! To me, this means that Alice's world (of abelian groups) is more general than Bob's world (of modules).

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would have been true whether I had stated it or not, but it is exactly the same, with the extra axiom actually adding nothing whatever to the concept. It would be totally unhelpful if we were to use the word "generalisation" in such a way that something is a generalisation of something else if we define it in one way, but not if we define

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gave in answer to that opening post: "A concept A is a generalization of a concept B when B is a special case of A" or, equivalently, A concept A is a generalization of a concept B when every example of B is an example of A, but not necessarily vice versa. In the sense which D.Lazard then went on to

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I disagree with the idea that all groups and rings are generalizations of sets. They are sets with additional axioms imposed on them, so evidently they are not generalizations, and I would not really call them "special cases" of sets either because a group, for example, is not just a set, but a tuple

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The notion of module is a generalisation of that of abelian group because (as explained by D. Lazard) every abelian group has a structure of a module (of the ring ZZ) which allows to recover its structure as an abelian group. The converse is clearly not true (for instance a free abelian group of rank

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EDIT: It took me a while but I finally understood what you meant. If we were to treat abelian groups as Z-modules in the way that you described, then in that case YES, modules WOULD be a generalization of abelian groups (instead of having the set Z and a particular function, you have a general ring R

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That doesn't matter: it has a scalar multiplication; that scalar multiplication is not included in the axioms because it follows naturally from the existing axioms, not because it isn't there. Suppose I choose to define something, let's call it a JBW group, with exactly the same axioms as an Abelian

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On the other hand, the "forgetful" construction that you mention can be useful only insofar as you are interested in properties of modules that follow from their mere additive structure. To illustrate further why your conception of "generalisation" is fraught, let me say that all groups, rings, etc.

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An abelian group is a generalization of an R-module because every example of an R-module induces an abelian group in a natural way but not necessarily vice versa. Consider, for instance, an F-module where F is the set of real numbers and the module is F^n (the n dimensional euclidean vector space).

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abelian groups and sets, exactly as do rings and fields. Similarly, all are sets. "Generalizing" does not mean dropping some axioms. It means enlarging the context in a way that allow applying some properties to new situations. Here, almost all important properties of abelian groups remain true for

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has expressed the principle that "When either of two styles are acceptable it is inappropriate for a Knowledge editor to change from one style to another unless there is some substantial reason for the change." If you believe an alternative style would be more appropriate for a particular article,

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In a nutshell, a semimodule is just like a module, except that the underlying abelian group is replaced with an abelian semigroup, so the elements do not necessarily have inverses. For example, the set of natural numbers is a semiring, and it is a semimodule over itself just as any ring is a module

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it relaxes the set of solutions big time, and i can't see how the properties we appreciate for positive-definiteness simply carry over to the former "just because" we extended the solution set to include zero, which means any psd matrix is NECESSARILY NOT pd because its distinguishing attribute is

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EDIT: I will add that many mathematicians attempt to generalize something as much as they can, and often by generalizing something, you end up with more stuff to do. But this isn't always the case. Sometimes if you drop some axiom, you actually lose very interesting results, and sometimes having a

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Do you, or do you not, agree with the idea that A cannot generalize B while B already generalizes A? If you agree with that idea, then you can't have an additional remark like "in any useful sense". If abelian groups generalize R-modules in any sense whatsoever then we can deduce that R-modules do

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A concept A is a generalization of a concept B when B is a special case of A. Here, the abelian groups are exactly the modules over the integers, and the vector spaces are exactly the modules over a field. So, it is correct to say that vector spaces and abelian groups are special cases of modules,

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What's actually going on is that any abelian group can be blessed with a useful scalar multiplication operation, and when you do that, the axioms of a module are satisfied. To me, that does not mean that modules generalize abelian groups, but rather that there is a useful construction you can make

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terms, and ... ". It logically follows from SchwaWolf's position that a module is a generalisation of what I have defined, as I have added an extra axiom. However, what I have defined is simply the concept of "Abelian group"; I have done so in an unnecessarily verbose way, stating something which

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Modules generalise abelian groups because the former allow to deal with many more objects. D. Lazard's example of the structure theorem for f.g. modules over PIDs is an excellent illustration of why this is useful: it shows that the concept of module allows you to fit the structure theorem for fg

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If we have two people Alice and Bob living in very restricted logical worlds where Alice's world is governed by the axioms of abelian groups and Bob's world is governed by the axioms of R-modules, then every true statement that Alice comes up with will correspond very naturally to a statement in

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and "texhtml" encoding may have different presentations for registered users, depending on user preferences and personal styles." This would suggest that mvar is not among the top 3 choices for encoding. There is a LOT of rendering variability which occurs when we use multiple templates, whereas

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For those links, where do you see the language which suggests there is consensus to use math and mvar? There is also language to suggest that encoding and typography should be locally decided, rather than decided on a Knowledge math-wide level. We also have this statement: "This essay offers a

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If they do (the argument being that "there are more jury modules than modules" since there may be several well-established juries and we may agree that if f(m)=6 for all m in M then M is a jury module automatically, meaning every module induces a jury module in an obvious way but there may be

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Z where Z is the set of integers such that for every m in M, f(m) maps to the decision of a well-established jury (an integer from 1 to 10) where the jury decided how appealing that particular element is. If the module is infinite then the default rating is 6 and the jury decides their rating

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This is a question of whether UNDERNEATH the text there will be one kind of element tag versus another, and how that impacts site accessibility. I'm here to argue why there are net positive reasons for local style consistency on this page, but so far I've been getting no response to my actual

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If they don't, then I urge you to think about how this jury function is any different than imposing a scalar multiplication function (and a set of scalars) for some abelian group. After all, according to you, the action of doing that generalizes abelian groups. So why should jury modules not

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started this discussion by saying "Usually when we generalize something, we drop unneeded axioms", which is of course perfectly true, but that is just a property of how we commonly define generalisations, not a definition of what "generalisation" means. A correct definition is the one which

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If you define integers as a certain kind of principal ideal domain then I agree that principal ideal domains generalize integers because then, a principal ideal domain is just the same requirements that you'd require for the integers except you don't require that the positive elements are

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Also you discuss on the meaning of "generalization" as if it would be a mathematical concept subject to a formal definition. This is not the case here. The word is herefor informally explain the relatioship between the topic and other related concepts. In other words, it is used here as

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forms a sequence of algebraic structures where each structure (more or less) contains the same axioms as the previous one plus some additional axioms. Therefore they are sorted from most general to least general, and in particular abelian groups generalize modules, not the other way

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Improved site accessibility for screen readers, reduced variability of rendering due to browser interactions, massive technological support and great fallback solutions, etc. are all well known arguments about why there is non-trivial BENEFIT. Where does focus on style fit in all of

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Of course one has to be careful with what equality really means, since in a pedantic sense you could argue that the natural number 1 is not the same as the real number 1 on the basis that they have different set-theoretic constructions. But that's obviously not a helpful notion of

795:(3) It is not obvious why there will be edit wars over a matter of encoding or rendering, especially as the rest of the world outside of Knowledge has very strong consensus for Latex and MathML. Is there actually a problem of people converting Latex into a myriad of math templates? 1251:

I don't understand how you're simultaneously claiming that my reductio ad absurdum view of generalizations is both fraud, and yet you're using the exact same notion to claim that modules generalize abelian groups. Could you try to enlighten me on where the analogy breaks down?

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Now elements of this set form an abelian group in a natural sense, but what is the natural way to extend the abelian group F^n into an R-module for some ring R? It could be the vector space (F, F^n) or it could be the "spiced up abelian group" (Z, F^n) where Z are integers.

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How is that different from saying that a vector space generalizes R-modules because we enlarge the context in a way that we can apply some properties to new situations, like for example we can now invert scalars and thus show that each basis has equal cardinality?

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To really emphasize why modules contain more axioms than abelian groups, consider that an abelian group imposes the existence of ONE set with certain criteria, while a module imposes the existence of TWO sets (one of which having the same abelian group axioms).

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Any theorem about positive definite matrices carries over seamlessly to a statement about positive semi-definite matrices, so therefore positive definite is the more general setting, despite positive semi-definite matrices being the larger set with respect to

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1. Usage of multiple templates or raw unicode leads to a LOT of rendering variability and browser interaction, whereas Latex focuses efforts of mathematical typography and rendering onto preferred community libraries. This makes usability very hard to test.

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There's lots of possibilities in this direction, because you're ADDING axioms and you have different ways to satisfy those new axioms. When you generalize something, you drop some axioms, and the resulting structure is highly obvious, which is not the case

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that is useful for explaining the context. So, the common use of the word by mathematicians is more important than a formal definition. If you don't like this use, replace it in your mind with "strongly related", this will do the job.

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So if abelian groups generalize R-modules, then in particular it means that R-modules do NOT generalize abelian groups. That's certainly not a useless result as is evident by the length of this comment thread.

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You note that a free abelian group of rank 2 has infinitely many non-isomorphic modules associated with it but if anything, doesn't that just solidify the point that an abelian group is a more general set of

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An abelian group does not have the structure of a module because an abelian group does not take the existence of scalar multiplication (let alone the existence of a distinct set of scalars) as an axiom.

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This is easily explained : i made a mistake while typing my answer, your contention regarding groups vs. sets would be that sets are generalisations of groups (not the reverse). The rest stands.

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Let us consider a world where we have a new algebraic structure "jury module" M that is in all other respects a module, except we additionally require that there exists some function f : M -: -->

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also containing an operation defined over the elements of that set. In particular, it's not a set with some of the set theory axioms dropped out, which would in my view make it a generalization.

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i'm tired of these physicists abusing numbers and equations under the guise of mathematics. stop calling positive-semidefinite ANYTHING a generalisation of positive-definiteness. it's not!!

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i want to chime in here as well, because i am tired of people in "physics" acting like their gymnastics qualify as mathematics. analogous to the discussion above is this belief that semi-

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Here, on the other hand, I don't think it's pedantic at all to say that a set and a group are two different things and neither equals the other, I think that's just common sense.

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It is true that every module is an abelian group. But one has also that every abelian group is a module over the integers, with the scalar multiplication defined naturally by

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different ways to turn a module into a jury module) then I argue that the word "generalize" has completely lost its meaning and we are not on firm mathematical ground anymore.

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So, one passes from abelian groups to modules by dropping the requirement that the ring is the ring of integers. A witness that this is a relevant generaliztion is that the

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In the second paragraph the term "abelian" should be linked to our relevant article for ease of use. I'm not a skilled editor, so request someone else add this link. Thx

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4. Maintenance becomes easier with uniformity. Across different math pages one may find raw unicode, a no-wrap template, a variables template, or a generic Math template.

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explain, the concept "module" is clearly a generalization of the concept "Abelian group". Endlessly trying to argue that that is not the case is really not constructive.

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before implementation. If there is no positive response, or if planned changes affect more than one article, consider notifying an appropriate Wikiproject, such as

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a set. An abelian group, to me, is a tuple of both sets and operations, while a set is just a set. So how could they be the same thing in any reasonable sense?

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are "generalisations" of sets as they are just sets with added structure: but this will not get you far if you want to prove results about groups or rings.

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But certainly either A generalizes B or B generalizes A or neither. Certainly it can't be that A generalizes B and B generalizes A, or what do you think?

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says not to proceed BOLDLY and to build consensus on the talk page. That suggests that local talk pages are precisely the right place to build consensus.

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Despite being a slightly off-topic rant, I found your comment interesting and I agree with you entirely! Thank you for the good thought-provoking post.

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I disagree with this statement. If it's not included in the axioms, then from the point of view of comparing different sets of axioms, it isn't there.

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I mean that "abelian groups generalise R-modules" is a completely useless proposition, as is "sets generalise groups". Sorry for not being clearer.

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An abelian group does not have the structure of a module because an abelian group does not take the existence of scalar multiplication as an axiom.

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Unfortunately I can't comment on the structure theorem nor a category theory approach because that is beyond the scope of my current understanding.

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Of course, this "generalization" is rather superficial, since of course dropping the scalar multiplication is hardly a generalization at all.

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that satisfies the following constraints: ...". Knowledge must follow the common usage of mathematicians rather than your own thoughts.

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3. It's easier for amateur Wikipedians to copy-paste Latex, and it's easier to follow along when the community has consistent style.

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Okay... wouldn't that mean that abelian groups generalize R-modules because an R-module contains more axioms than an abelian group?

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from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. Proposed changes should generally be discussed on the

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Large-scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting

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rendering with Latex is more battle-hardened and continuously improved upon. Some of the rendering is typographically AMBIGUOUS.

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abelian groups, and the classification of conjugacy classes of matrices (among many other results) under the same umbrella.

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comparison of different encodings and presentation of mathematical formulae. The three principal ones are the <math: -->

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The sentence "If K is a field" defines K. More precisely, it specifies K sufficiently for giving sense to what follows.

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discuss this at the article's talk page or—if it raises an issue of more general application or with the MoS itself—at

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Every R-module (less general) induces an abelian group (more general) but not every abelian group induces an R-module.

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SchwaWolf seems to have essentially two objections to the idea that "module" is a generalisation of "Abelian group":

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Sorry, what do you mean by "the second proposition"? Do we in fact agree that abelian groups generalize R-modules?

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Every vector space (less general) induces an R-module (more general) but not every R-module induces a vector space.

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However, there is a confusion of two senses of "more axioms" here. "Field" has more axioms than "Abelian group",

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generalise R-modules in any useful sense. The converse is true as illustrated by the example given by D. Lazard.

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Usually when we generalize something, we drop unneeded axioms and make something work in a more general setting.

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Wouldn't it therefore make sense to say that abelian groups generalize Z-modules, and not the other way around?

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The converse is somewhat true because a subset of the axioms of a module forms the axioms of an abelian group.

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However, when comparing modules and abelian groups, modules contain strictly more axioms than abelian groups.

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group, but with the additional requirement "There exists an operation called "scalar multiple" such that if

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Again, there is no point in making your case here. Suggested changes to the consensus should be discussed at

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Therefore I think modules generalize vector spaces while abelian groups (trivially) generalize any R-module.

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You can probably express all of the above in categorical language if you fancy that kind of thing. Cheers,

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Proposition. Latex should be preferred as the default for mathematical typography on this page because:

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Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.

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To go from an abelian group to a module means to define an additional scalar multiplication operation.

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To go from a vector space to a module means to drop the requirement that scalars must be invertible.

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If people somehow managed to internalize this idea then the original debate would also be settled.

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over itself. I think an argument could be made to redirect this to "semiring" instead of "module."

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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6. Mathematical typography should be consistent at least within-page even if not between pages.

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would never say that a group is a set, but most mathematicians do. For example, the article

1404:. Similarly, principal ideal domains are generalizations of integers, in the sense that the 874: 576: 507: 402: 371: 201: 1548:

in the sense that it has an extra operation, and that extra operation is defined by axioms

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Modules contain more axioms than Abelian groups, but a generalisation has fewer axioms.

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So you think that sets generalize groups because groups contain more axioms than a set?

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Fundamental theorem of finitely generated modules over a principal ideal domain

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well, in this case, that fails per Horn and Johnson's Matrix Analysis (2013):

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There's something weird going on because I would never say that an R-module

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2. Latex provides many fallback options, including rendering to SVG or PNG.

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7. More popular peer math pages prefer this style, such as Linear Algebra.

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tag, raw wiki (or HTML) code, and "texhtml" templates. The <math: -->

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otherwise. If you want to change this consensus, please, discuss it on

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manually for some amount of elements, based on how they feel that day.

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but in the sense that there is an extra axiom placing a restriction

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and so modules generalize both vector spaces and abelian groups.

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and many other properties of integers can be extended to them.

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Is it reasonable to say that modules generalize abelian groups?

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5. Editing by source becomes very ugly with multiple styles.

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for every abelian group that turns the group into a module.

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2 has infinitely many non-isomorphic ZZ-module structures).

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The examples refer to K but that is not defined anywhere.

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Edit-warring over style, or enforcing optional style in a

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Fundamental theorem of finitely generated abelian groups

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The word redirects here but is not used in the article.

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It would. That's why the second proposition is useless.

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and to revocation of semi-automated tools privileges.

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Thanks. 174: 8: 1133:(almost the same statemrnt and same proof). 1713: 1572:is any integer, then the scalar multiple 1430:an abelian group or that an abelian group 457: 256: 1152:Do these jury modules generalize modules? 1086: 1037: 1769:https://math.stackexchange.com/a/2156221 755:Knowledge:AutoWikiBrowser § Rules of use 1798:Knowledge vital articles in Mathematics 1761: 1712:also one that introduces singularity!! 734:was invoked but never defined (see the 696: 258: 217: 657: 640: 619: 1813:B-Class vital articles in Mathematics 800:Knowledge:Manual of Style/Mathematics 787:Knowledge:Manual of Style/Mathematics 7: 1568:is any element of the JBW group and 304:This article is within the scope of 726: 247:It is of interest to the following 23:for discussing improvements to the 1129:is a direct generalization of the 14: 1823:Top-priority mathematics articles 1406:fundamental theorem of arithmetic 798:(4) I refer to this passage from 662:fashion without prior consensus, 324:Knowledge:WikiProject Mathematics 1793:Knowledge level-4 vital articles 370: 327:Template:WikiProject Mathematics 291: 281: 260: 227: 218: 45:Click here to start a new topic. 1684:is somehow a generalisation of 1376:not generalize abelian groups. 344:This article has been rated as 1803:B-Class level-4 vital articles 1097: 1088: 1066:{\displaystyle nx=x+\cdots +x} 648:Knowledge talk:Manual of Style 446:13:59, 11 September 2015 (UTC) 430:06:58, 11 September 2015 (UTC) 1: 785:I also refer to this passage 478:Proposition: Latex conversion 318:and see a list of open tasks. 42:Put new text under old text. 1818:B-Class mathematics articles 883:17:47, 22 January 2021 (UTC) 680:15:15, 22 January 2021 (UTC) 611:15:11, 22 January 2021 (UTC) 585:14:21, 22 January 2021 (UTC) 516:14:35, 22 January 2021 (UTC) 897:Examples - K is not defined 525:. 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Welcome! 1839: 1005:To put it yet another way, 632:for mathematical articles. 527:WP:WikiProject Mathematics 1751:21:34, 23 June 2024 (UTC) 1728:03:17, 18 June 2024 (UTC) 1673:14:02, 16 June 2024 (UTC) 1637:18:53, 13 June 2024 (UTC) 1554:on the existing operation 1504:15:02, 12 June 2024 (UTC) 1458:14:21, 12 June 2024 (UTC) 1386:14:23, 12 June 2024 (UTC) 1118:{\displaystyle (-1)x=-x.} 807:Help:Displaying a formula 472:23:12, 14 June 2016 (UTC) 343: 276: 255: 80:Be welcoming to newcomers 1418:14:55, 7 June 2024 (UTC) 1371:14:03, 7 June 2024 (UTC) 1353:12:31, 7 June 2024 (UTC) 1335:08:19, 7 June 2024 (UTC) 1321:08:14, 7 June 2024 (UTC) 1307:14:12, 6 June 2024 (UTC) 1293:13:13, 6 June 2024 (UTC) 1276:16:19, 4 June 2024 (UTC) 1262:13:10, 4 June 2024 (UTC) 1230:07:17, 2 June 2024 (UTC) 1204:15:45, 1 June 2024 (UTC) 1143:15:14, 1 June 2024 (UTC) 1028:14:39, 1 June 2024 (UTC) 986:21:03, 30 May 2024 (UTC) 970:19:31, 30 May 2024 (UTC) 937:13:35, 3 July 2021 (UTC) 921:13:13, 3 July 2021 (UTC) 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