Talk:Module (mathematics)

What is a semimodule?

The word redirects here but is not used in the article. Equinox (talk) 06:58, 11 September 2015 (UTC)

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 over itself. I think an argument could be made to redirect this to "semiring" instead of "module." Rschwieb (talk) 13:59, 11 September 2015 (UTC)

Add a Link Please

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 — Preceding unsigned comment added by 97.125.86.137 (talk) 23:12, 14 June 2016 (UTC)

Proposition: Latex conversion

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

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.

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

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

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.

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

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

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

SirMeowMeow (talk) 14:35, 22 January 2021 (UTC)

This has been widely discussed in Wikipedia, see MOS:MATH. The current consensus on WP:WikiProject Mathematics is to use templates {{math}} and {{mvar}} for simple inlines formulas, and <math> otherwise. If you want to change this consensus, please, discuss it on WT:WPM.

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 Wikipedia math-wide level. We also have this statement: "This essay offers a comparison of different encodings and presentation of mathematical formulae. The three principal ones are the <math> tag, raw wiki (or HTML) code, and "texhtml" templates. The <math> 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 rendering with Latex is more battle-hardened and continuously improved upon. Some of the rendering is typographically AMBIGUOUS.

When you use something other than <math>, you are losing mass-browser support, huge amounts of accessibility work, and multiple-fallback solutions. Doing so ought be a deliberate choice made for conscious tradeoffs. SirMeowMeow (talk) 14:21, 22 January 2021 (UTC)

Again, there is no point in making your case here. Suggested changes to the consensus should be discussed at WT:WPM as suggested above. Rschwieb (talk) 15:11, 22 January 2021 (UTC)

MOS:FORMULA says Large-scale formatting changes to an article or group of articles are likely to be controversial. One should not change formatting boldly from LaTeX to HTML, nor from non-LaTeX to LaTeX without a clear improvement. 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.

MOS:VAR says The Arbitration Committee has expressed the principle that "When either of two styles are [sic] acceptable it is inappropriate for a Wikipedia 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, discuss this at the article's talk page or—if it raises an issue of more general application or with the MoS itself—at Wikipedia talk:Manual of Style.

and

Edit-warring over style, or enforcing optional style in a bot-like fashion without prior consensus, is never acceptable.

So, please, respect wikipedia rules. Thanks. D.Lazard (talk) 15:15, 22 January 2021 (UTC)

References

(1) MOS:FORMULA 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.

I also refer to this passage Wikipedia:Manual of Style/Mathematics, 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."

(2) The citation to the Wikipedia 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?

(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 Wikipedia has very strong consensus for Latex and MathML. Is there actually a problem of people converting Latex into a myriad of math templates?

(4) I refer to this passage from Wikipedia:Manual of Style/Mathematics: "Even for simple formulae the LaTeX markup might be preferred if required for uniformity within an article."

(5) I refer to the page Help:Displaying a formula, which is almost entirely full of Latex and MathML examples. This was the start of many people's on-ramping for learning to encode math on Wikipedia.

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 arguments.

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 this?

SirMeowMeow (talk) 17:47, 22 January 2021 (UTC)

Examples - K is not defined

The examples refer to K but that is not defined anywhere. — Preceding unsigned comment added by Gcsfred2 (talk • contribs) 13:13, 3 July 2021 (UTC)

The sentence "If K is a field" defines K. More precisely, it specifies K sufficiently for giving sense to what follows. D.Lazard (talk) 13:35, 3 July 2021 (UTC)

Is it reasonable to say that modules generalize abelian groups?

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

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

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

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

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 for every abelian group that turns the group into a module. SchwaWolf (talk) 19:31, 30 May 2024 (UTC)

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, and so modules generalize both vector spaces and abelian groups. D.Lazard (talk) 21:03, 30 May 2024 (UTC)

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

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

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

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

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

To put it yet another way,

abelian group -> module -> vector space

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 around.

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). SchwaWolf (talk) 14:39, 1 June 2024 (UTC)

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 $nx=x+\cdots +x$ (n summands) for positive n, and $(-1)x=-x.$ 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 Fundamental theorem of finitely generated modules over a principal ideal domain is a direct generalization of the Fundamental theorem of finitely generated abelian groups (almost the same statemrnt and same proof). D.Lazard (talk) 15:14, 1 June 2024 (UTC)

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

Do these jury modules generalize modules?

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 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.

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 generalize modules?

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 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).

SchwaWolf (talk) 15:45, 1 June 2024 (UTC)

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

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

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. 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.

You can probably express all of the above in categorical language if you fancy that kind of thing. Cheers, jraimbau (talk) 07:17, 2 June 2024 (UTC)