|Module (mathematics) has been listed as a level-4 vital article in Mathematics. If you can improve it, please do. This article has been rated as B-Class.|
|WikiProject Mathematics||(Rated B-class, Top-importance)|
- 1 Two questions
- 2 ??
- 3 bimodules over commutative rings
- 4 Module (category theory)
- 5 Modules are not modular
- 6 faithful module
- 7 lead section
- 8 fomal should be formal
- 9 Motivation
- 10 Easy Question
- 11 Semimodules
- 12 Modules generalize both vector spaces and ideals
- 13 on the other hand
- 14 Missing essential examples
- 15 Scalar multiplication?
1) In the fifth example, it is written
- the category of C∞(X)-modules and the category of vector bundles over X are equivalent.
Is this actually true? I think it should be the category of finite-dimensional projective C-infinity modules, based on the Swan's theorem page.
2) What is a pseudo-module, in the parlance of Bourbaki? This is what I glanced at the article to look up in the first place. (I tried a google search to no avail. Comp sci usage and other mathematical usage overwhelms whatever meaning might be there.)
Thanks. 188.8.131.52 07:29, 1 December 2006 (UTC)
- Bourbaki takes all rings to have identity and the identity to act trivially on all modules. A pseudo-module over a pseudo-ring (ring without identity) is what is called a module here. Changing conventions can be somewhat annoying... 184.108.40.206 01:02, 3 March 2007 (UTC)
All ring modules are not unital. --S. A. G.
- I think above refers to a problem with the article, that I think is fixed ?? linas 00:38, 6 Apr 2005 (UTC)
I agree, and the glossary of ring theory does not even stipulate (which is a good thing!) that a ring must be unital (i.e. a unit ring). But this is somehow an eternal debate... — MFH: Talk 21:03, 12 May 2005 (UTC)
bimodules over commutative rings
I'm not so sure if the statement
If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.
is correct, or at least, not misleading. I think one can consider a bimodule where the left and right action of the ring are completely different operations. (I think people working on Hopf groups, e.g. in noncommutative geometry, consider such things, but there seems to be no info on this here.) — MFH: Talk 21:03, 12 May 2005 (UTC)
- The statement you cite is correct, but it has little to do with bimodules. Bimodules require the two module actions to commute, and there are examples of modules which are left R modules and right S modules, but no bimodule structure is possible. Every R-S bimodule structure on M corresponds to an R⊗ZS module structure on M. And of course, every bimodule is independently a module over both rings.
Hi there! Does the concept of Module (category theory) make any sense to you? If so, would anybody write an article about this? This article seems to be requested on Wikipedia:Requested articles/mathematics. I suspect the person requesting this article confused something, but I could be wrong. Thanks. Oleg Alexandrov 00:15, 29 Dec 2004 (UTC)
- Hmm. Here, this gives a rich treatment: module in nLab. So currently, Module (category theory) is a redirect to this article, but probably should not be; it should look more like the nlab article. However, if this article were created today, it would be an orphan. Oh well... linas (talk) 15:17, 2 September 2012 (UTC)
- Nothing about that link really justifies making "module (category theory)" its own article. Really, it's more often that you learn about modules, then later that they are an example of a category. If you get deep into category theory you can generalize modules more (as this link does) and do advanced categoric things with them, but I think that's above and beyond what normal students would encounter. Rschwieb (talk) 00:24, 3 September 2012 (UTC)
Modules are not modular
The redirection Modular -> Module -> Module (mathematics) is nonsense because there is no module corresponding to modular forms (and related concepts), nor in modular representation theory.--220.127.116.11 14:18, 28 Feb 2005 (UTC)
- I fixed the disambig links for the above complaint. linas 17:14, 12 Mar 2005 (UTC)
"Faithful. A faithful module M is one where the action of each r in R gives an injective map M→M. Equivalently, the annihilator of M is the zero ideal." This is incorrect, a faithful module is one where the action of each nonzero r in R is not the trivial map, i.e. if a does not equal b in R, then a->ax and b->bx are different endomorphisms. These statements are equivalent to ann(M)=0.
I think this article (like all articles) needs a good lead section, and I think the current
- In abstract algebra, the notion of a module over a ring is the common generalization of two of the most important notions in algebra, vector space, and abelian group.
is doing a bad job. It doesn't actually define what a module is, the abstract algebra classification makes modules sound vastly less important than they actually are (it sounds like it's some exotic structure of interest mostly in universal algebra, like a magma actually is (apologies to anyone deeply in love with magmas, but they do seem exotic right now)).
I do not see how module is the common generalisation of vector spaces (F-modules, where F is a field) and abelian groups (Z-modules); both of those are actually, in modern terminology, R-R-bimodules, and that would be their common generalisation: R-R-bimodules where R is a commutative domain. Going all the way to real modules is quite a bit more general; noncommutative rings, in particular, are important.
(I'm also curious about vector spaces being such an important notion - instructive, maybe, but nowadays you just think of them as modules over a field.)
I'd suggest something along the lines of:
- In mathematics, a module over a ring is a space whose elements can be added and multiplied by elements of the ring, analogously to vector spaces. Formally, a module M over R is an abelian group with a ring homomorphism R → Hom(M, M).
But that doesn't sound quite good enough for me to be bold without asking for better proposals first.
RandomP 19:47, 29 April 2006 (UTC)
- I like the current intro more however, it reads better, is more elementary and more motivational than what you suggest I think. Maybe you can add the sentences you want under the current intro rather than replacing it? Oleg Alexandrov (talk) 20:23, 29 April 2006 (UTC)
- The audience of this article includes people with only the barest knowledge of abstract algebra, and I think for these people the current introduction provides more motivation. There's nothing stopping the more formal, technical definition from appearing lower down. Certainly there should be no mention of Hom in the introduction, it's unnecessarily technical. On the other hand, I do agree that the statement that modules are a common generalisation of vector spaces and abelian groups is a bit misleading. I noticed this some time ago and wasn't quite sure how to fix it. Perhaps it's fair to say that "module" is a generalisation of "vector space", but not so fair to call it a generalisation of "abelian group". That would after all be much the same as saying that a vector space is a generalisation of an abelian group, which sounds a bit silly. Rather I would say that a module is an abelian group with additional structure. Dmharvey 20:59, 29 April 2006 (UTC)
- Yes, but it is by no means the simplest way to generalise both. Furthermore, abelian groups are rarely thought of as having an external multiplication with Z until after they're proved to be Z-modules. I'd suggest saying they also generalise vector spaces over skew fields, but those are introduced rarely, I fear.
- In any case, the blatant bias towards the commutative case needs to go!
- RandomP 21:28, 29 April 2006 (UTC)
- Edit conflict: We were talking about this issue at talk:manifold last week. Dmharvey is right in general, it doesn't have much meaning to call something with extra structure a generalization of something with less structure. In general, it really only makes sense to talk about things which have fewer axioms (but the same size structure) a generalization. Thus, monoids are generalizations of groups, topological manifolds are generalizations of smooth manifolds, and Banach spaces are generalizations of Euclidean space, but rings are not generalizations of groups (rings have more structure, not every group can be a ring), manifolds are not generalizations of topological spaces (not every space can be a manifold), and Banach algebras are not generalizations of Euclidean space. But Oleg is also right: in this instance, every abelian group is a module. The extra structure that modules have is somehow degenerate in the case of Z-modules. I wonder what the technical explanation is for what's going on there. Nevertheless, for arbitrary modules, that extra structure is nontrivial, and therefore it may not make much sense to call a module a generalization of an abelian group. Even if it does make sense, is it useful in practice? Does it merit mention in the intro? -lethe talk + 21:29, 29 April 2006 (UTC)
- I'd definitely not want to say that "a module generalizes ... abelian groups", for the same reason Dmharvey stated. This is very misleading. It suffices to note that abelian groups "are the same things as" Z-modules. — merge 10:01, 30 April 2006 (UTC)
- It's definitely an improvement! However, now it suggests, rather misleadingly, that modules are important only for commutative algebra, which itself is important in a number of other fields. The noncommutative case is important, and ignoring it before even given the formal definition is more misleading than skipping it alltogether.
- What modules are useful for might change, but what modules are is not going to.
- RandomP 17:54, 30 April 2006 (UTC)
- Unfortunately I am quite ignorant when it comes to modules over non-commutative rings. The only such objects I ever deal with would be modules over a group ring such as Z[G], so I guess this is really just representation theory, which is mentioned in the introduction. If you can suggest other areas where modules over non-commutative rings come up, you're more than welcome to add them to the intro. Dmharvey 18:04, 30 April 2006 (UTC)
The point is that the representation theory of groups, for example, needs modules, not commutative algebra.
- Modules are one of the core notions of commutative algebra, which is essential in many important fields of mathematics, including algebraic geometry, homological algebra, algebraic topology, and the representation theory of groups.
suggests that all modules are good for is commutative algebra.
RandomP 18:57, 30 April 2006 (UTC)
FWIW, I'm not usually being bold with the lead section precisely because others might prefer the way it currently stands. Now there appears to be some consensus that we need a new lead section, that's maybe not so much of an issue.
I still think that the important thing, and the thing the reader should actually gain from the article, is the definition. Yes, modules are important for pretty much everything even vaguely research-related (in pure mathematics and non-classical theoretical physics, at least) today, but really, knowing what a module is is more important than knowing what it's good for.
Currently, using a half-screen browser layout, the definition (and let's be clear here: the "formal" in there isn't really required) just barely still fits on the same page. A linear reader will stumble over all the complicated scary terms, and quite possibly give up on the article, before even being told what a module is.
I'm not the ideal person to fix that; I'm just not a terribly good writer. But I can point out that it's wrong, and that I'd consider even the clumsy
- A module M over a ring R is an abelian group (M, +) together with a multiplication operation R × M → M.
better: it's a complete definition, since the term "multiplication" implies distributivity and associativity.
I can live with a non-scary sentence about that really being the same thing as a vector space preceding that; the current best practice appears to be that fields and vector spaces come first, then rings and modules, so no reader who's scared by vector spaces is going to profit much from learning about modules (I don't consider this a general problem, it's more of a deficiency in the literature). But the definition is the important bit, and if there's no good way to say it without using symbols, well, we'll just have to use symbols instead.
RandomP 20:18, 30 April 2006 (UTC)
- Agree that "formal" is not required.
- Disagree that you're not a good writer. You seem eminently capable of stringing words together.
- Agree that the "definition" section is too long. Some of that material should be put in a separate section.
- The trick here is to be serving two audiences at the same time. On the one hand we want the concise version "an R-module is a ring homomorphism from R to End M for an abelian group M", and on the other hand we want something accessible to someone learning this kind of algebra for the first time. I'm actually thinking in this case that it might be okay to put the "advanced" version first, as long as it's very brief, and then immediately following that have something like "Explicitly, this means the following:...", and spell out the "multiplication map", with the properties that it needs to have, as we have listed currently. Usually, I would insist on the baby version up front, but here it seems like it might not be such a problem. Dmharvey 20:41, 30 April 2006 (UTC)
- Here's an attempt with a symbol-free version, with the baby version kept as "formal definition". I really think this might be the main problem; there just isn't a good English-language description of modules.
- what a module really is, I'd say, is a thingy that a ring acts on. Of course that kind of requires it to have an addition, and thus be an abelian group, but you don't really think of it that way. Or do you?
- RandomP 21:07, 30 April 2006 (UTC)
Perhaps modules generalize ideals? 18.104.22.168 01:06, 3 March 2007 (UTC)
The "which are .." clause in the second sentence: "Modules also generalize the notion of abelian groups, which are modules over the ring of integers." is confusing in that a representation of a finite abelian group as a module over Z would require the ring of integers Z to be wrapped upon itself in the operation Z × M → M to keep M finite. If I were required to represent a finite abelian group as a module, I would prefer to use a finite ring of the smallest size (like Z2) that produced whatever additional module properties might be required. I see no advantage to such a representation over the direct product of cyclic groups of orders prime to some power. I agree with RandomP and Dmharvey that modules do not generalize abelian groups. You might say they extend abelian groups, adding structure. Saying that abelian groups "are" modules just seems wrong. If no one objects, I would like to change this second sentence to read: "Modules extend the notion of abelian groups, adding a scalar multiplication by elements of a ring." Howard McCay (talk) 18:43, 26 November 2012 (UTC)
- I don't think you should remove the information, because it is a standard, straightforward description of how module theory is an extension of Abelian group theory. There is nothing strange about the existence of a representation even when M is finite. I could be misunderstanding. Could you clarify what it is that you are aiming to change? Rschwieb (talk) 01:59, 27 November 2012 (UTC)
fomal should be formal
fomal should be formal
- Thanks for pointing that out! It might have saved you some time to be bold and edit it yourself, but pointing out problems on the talk page is perfectly fine, too, of course.
- RandomP 21:21, 14 May 2006 (UTC)
I think the motivation section could be improved. I like what is there now but I think a few sentences about the "WHY" in the motivation section are warranted. The current paragraph is misleading in that it suggests that a desire for mere generality is the primary motivation for modules. There are numerous objects in mathematics which represent generalizations which are not particularly useful of interesting. Modules are a very rich mathematical object that have proved both very interesting as an abstract object of study, and also very useful in terms of applications to problems arising in various branches of mathematics. For example, the theory of modules can be used to study the structure of a linear operator over a vector space, and an understanding of modules is important for many areas within algebraic geometry, algebraic topology, etc. I think we would do well to include a little bit more of this explanation, even if it's just a couple sentences. On the other hand I don't feel like I know enough about these applications to be the one to include this text so I think someone else could do a better job than I! Cazort 14:51, 4 August 2007 (UTC)
At the beginning a module is defined as being over a ring instead of a field, but the first example says that the k-vector field is the same as the k-module. So as an ameteur, my distinction of what a module is, disappears...22.214.171.124 (talk) 15:52, 6 December 2007 (UTC)
- A field is a special case of a ring. That is why the intro says that the notion of a module is a generalisation of the notion of a vector space. It follows that an obvious example of a module is a vector space, but there are examples which are not vecotr spaces, because they are not over fields. How could the article make this clearer? JPD (talk) 16:26, 6 December 2007 (UTC)
I believe the section on Semimodules is not correct. In the literature there are two conflicting definitions of semimodules
- Wieland, Redei: commutative, cancellable ( regular ), unital semigroups
- modules over semirings
It is claimed that modules over semirings are just commutative monoids. Indeed the additive operation in semimodules is commutative and unital, but this property is common among all modules and can not be used to distinguish semimodules.
- The comment was not meant to classify semimodules, only indicate a difference between them and modules over rings. I have reworded it to make this clear. JackSchmidt (talk) 15:18, 20 February 2008 (UTC)
Modules generalize both vector spaces and ideals
Perhaps it could be emphasized, as in Atiyah-MacDonald, that modules generalize not only vector spaces but also ideals? That is a significant motivation for the concept. Typometer (talk) —Preceding undated comment was added at 20:22, 11 September 2008 (UTC).
- This is now mentioned in the motivation section. It was already mentioned in the examples section. JackSchmidt (talk) 20:43, 11 September 2008 (UTC)
on the other hand
What if we define our module over a rng (algebra) instead of a ring? What does wikipedia call this, and can it be linked somewhere prominently? That is, what is the name of a non-unital module. —Preceding unsigned comment added by 126.96.36.199 (talk) 23:22, 22 April 2009 (UTC)
- In the places they are studied they are just called "modules". I believe you'll find something on such stuff if you dig through some of Jacobson's stuff on rings without unity. Rschwieb (talk) 16:41, 14 August 2012 (UTC)
Missing essential examples
Two of the most important proto-examples are missing from this page: Modules over F[X] (which lead to Jordan's decomposition) and modules over group algebras (representations). 188.8.131.52 (talk) 19:15, 28 January 2012 (UTC)
I wonder why the article suggests/emphasises, at several points, that the ring operates by "scalar multiplication". This seems rather misleading. At the same time, one of the examples shows that a vector space is a module over the ring of its linear transformations. Bas Michielsen (talk) 16:31, 14 August 2012 (UTC)
- I think I can understand your reservations, but it seems to be a useful and mostly valid analogy with vector spaces. I can especially see your objection when there may be more than one type of "scalar". However here I think it's more helpful than it is misleading. It would be more confusing to use just "multiplication" to talk about the action of the ring on the module, and more clunky to write out "multiplication between a ring and module element". I removed two instances of "scalar" that I thought we could do without. If you think of a good workaround for 'scalar', let us know. Rschwieb (talk) 16:40, 14 August 2012 (UTC)