Talk:Free module

Finite linear combination

I commented in this article (words to the effect) that for a module with an infinite free basis, every element is a convergent series of multiples of the basis elements.

This has recently been changed to state that it is a finite sum of multiples of the the basis elements.

Is this change correct?

-- Onebyone 23:56, 28 May 2004 (UTC)

Well, I guess we could define a topology on the module, but unless that is the case, we can't define convergence. Phys 05:34, 5 Dec 2004 (UTC)

The change is correct, even for topological modules. --129.70.14.127 00:03, 8 November 2007 (UTC)

Yes. There are unhappily two conflicting meanings of the word "basis" in use, as regards e.g. a Hilbert space, or other topological modules. However, when deciding whether or not a module is free, the ordinary meaning (where we only consider finite linear combinations) is the one employed.

A related problem in the article is the fact that generating set now is linked to finitely-generated module, which of course is not very appropriate for an infinite basis. JoergenB (talk) 18:50, 22 August 2008 (UTC)

Definition

What is a "free basis"? Shouldn't it just be "basis"?--129.70.14.127 23:58, 7 November 2007 (UTC)

Done. --Beroal (talk) 15:35, 30 November 2010 (UTC)

Bad Introduction

I'm sorry to say that this article had a poor introduction. A free module is just a module that has a basis. Why start off talking about catagories and the like? It doesn't make sense. Nobody learns about vector spaces and modules from the catagory setting first. Everyone learns about the objects themselves, and once familiar we start to make our thinking more sophisticated by considering the bigger picture. This article was, for me, not at all usefull (even though I already knew what a free module was!) Dharma6662000 (talk) 21:06, 26 July 2008 (UTC)

Tere is one argument in favour of a category-theoretical introduction. In this way, the attribute "free" is put into its commonest(?) modern context from the beginning. However, there are clearly arguments against, too.-JoergenB (talk) 18:40, 22 August 2008 (UTC)

I would like strongly to second the opinion voiced by Dharma6662000, extend the argument at which that post hints, and counter the one argument offered by JoergenB. With apologies, I will be alluding to certain other math articles as well; I have two excuses: first, many of the same people contributing to these other articles may be referring to this article; and second, I don't know of any specific one forum that is obviously more appropriate for venting a general complaint.

Let us call a topic “foundational” if it bridges the mathematics that is typically familiar to scientists and engineers (or undergrads) with the more modern topics and formulations (such as those that a typical math student, say in English-speaking countries, might not encounter until math graduate school). The property of “being a foundational topic” of course admits of a degree, and is somewhat a matter of taste, not to mention a matter contingent on idiosyncrasies in one's math education. Nevertheless, some topics clearly qualify as being foundational, e.g. tensor products, exterior algebras, tangent bundles, and, I think, even free modules. (Other topics presumably are clearly not foundational.)

Who is the most likely reader of an article on a “foundational” topic? It is not someone who is already initiated into category theory, at least not if they intend to learn something new. For in order to learn category theory in the first place, they had to have already learned about the objects covered in the article (as Dharma6662000 explained perfectly).

This brings me to JoergenB's argument, as follows: why would one want to put “things in modern context from the beginning,″ given the near certainty that this will render the article nearly useless to the uninitiated? The only plausible answer I can think of is this: the already-initiated might read an article on a foundational topic for a sort of aesthetic reason. They might like having Wikipedia (to borrow a phrase from Tolkien) “filled with things that they already knew, set out fair and square with no contradictions,″ and in accordance with the sensibilities prevailing among the cognoscenti.

I think the price for having such articles is a much-reduced usefulness to the people who need these articles the most, and I think this price is way too high. Please do not misunderstand me: the more sophisticated perspective should of course be covered in the article—but near the end, after the less abstract treatment. From the perspective of a likely reader of an article on a foundational topic, the insistence of a contributor on shoving the “abstract nonsense” in the reader's face practically from the first line is at best unhelpful, and at worst intimidating and irritating. Personally, it has already happened to me several times to have tried to look something up in Wikipedia, only to give up in frustration and look for guidance elsewhere—successfully, by the way, which confirms that the topic can be covered in a more pedestrian way than done in Wikipedia. Please, do not make a Wikipedia article understandable only to those who already know nearly everything that's in it.Reuqr (talk) 05:05, 27 February 2009 (UTC)

Introducing a free module via category theory seems ridiculous to me as well. Although, I can't say I am surprised to see it written this way -- category theory appears in excess within many Wikipedia math articles. I think the subset of category theorists who are bent on describing every known mathematical idea via category theory are also the subset which most enjoy creating & modifying content on Wikipedia. Adammanifold (talk) 11:56, 19 May 2009 (UTC)

How can you define a free module as being a free module? In the first line... I'm not an expert at math, but that just seems really really bad! Perhaps if this isn't a mistake it could be clarified somehow. —Preceding unsigned comment added by 152.1.205.203 (talk) 20:43, 27 December 2009 (UTC)

Many facts about arrows could be taught at the elementary school level, instead of the nutty 19th century arithmetic curriculum that is taught today. This could be followed up by more challenging exercises at the high-school and college level. The fact that there is a wide-spread revulsion to category theory is due to a failing of the educational system, and due to the fact that the current educational system insists on pounding arithmetic into kids heads, instead of math. Thus, the concept of abelian groups, then groups, and vector spaces are all introduced as if they were generalizations of arithmetic. If we taught arrows to children instead, most of the objections above would be moot.

In other words: to those objecting to category theory above: the problem is really that your education has failed you; this is not a fault of category theory, but rather the mindset of the entire academic system. Now that we actually *do* have a better way of explaining things, we should go for the new and better way, instead of sticking to the olde-fashioned approaches.

Err, well, uhm, maybe I have to eat my words; I see this article never contained more than a single sentence about free objects, so, yes, this is rather opaque. And worse, the article on free objects defines them as a generalization of free examples from many other fields (which is in fact how I learned them too), rather than as things in and of themselves, from first principles. So we've got a generic problem here on WP. I'm wondering if/how this could be fixed... linas (talk) 16:04, 2 September 2012 (UTC)

Some areas for improvement

The article does not mention:

• projective modules, which under suitable restrictions are the locally free modules
• free ideal rings, and the commutative specialization principal ideal rings
• ideal class groups (measuring the difference between projective ideals and free ideals)
• grothendieck groups (measuring the same for f.g. modules)
• free resolutions, or the more popular finite free resolutions
• freeness is not preserved under Morita equivalence, the property generated by freeness under Morita equivalence is again the notion of projective module
• skew fields, the rings over which every module is free

The article's treatment of rank is poor. Rank should be given its own section.

For the zero ring, or rings without identity, free modules are very poorly behaved. For nonzero rings with identity: If E is a set, then every direct sum of E regular modules is determined up to isomorphism by the cardinality of E. If E, F are two such sets, E is infinite, and E and F have distinct cardinalities, then R^(E) and R^(F) are not isomorphic. Hence R^(E) has rank |E| if E is infinite. For some rings, R^m and R^n can be isomorphic modules without m being equal to n. This cannot happen for commutative rings, which should probably be mentioned. (Rings where R^m = R^n implies m=n, have the IBN property (linked in the article), and m is called the rank of R^m.

For the zero ring, the definition of free basis is slightly incorrect as it confuses a set (with its elements indexed) with an "indexed set", more commonly called a function from E to M. For nonzero rings, distinct e in E map to distinct elements of the module, but not for the zero ring (where all unital modules are free and isomorphic). JackSchmidt (talk) 19:40, 22 August 2008 (UTC)

Hungerford has a reasonable discussion of free modules over rings without identity, FWIW. They are basically free (unital) modules over that universal unital ring containing the non-unital ring. JackSchmidt (talk) 19:21, 8 May 2009 (UTC)

I have written a small section that gives pointers to the relation with projective and flat modules, but I think that more properties you mention (i.e. projectives being "almost" locally free) shouldn't be on this page but on the page for projective modules (or on another, corresponding, specialised page). Konrad (talk) 06:02, 11 May 2012 (UTC)

module

show that Q is not a finitely generated Z-module —Preceding unsigned comment added by 218.248.65.82 (talk) 06:35, 11 June 2009 (UTC)

"Every vector space is free"

This claim in the introduction is linked to a source which says "vector space" will mean a module over the complex numbers. So, even if the general result is true, I think the link should be removed, and the result should be proven in the article. — Preceding unsigned comment added by 98.248.228.230 (talk) 09:20, 27 November 2011 (UTC)

Cartesian product as a module

I know that this is a pretty minor point, but I disagree with these edits to regard the cartesian product of objects as implying a module structure: it is defined as a set, and is defined for any sets. When being careful, one should also not refer to the direct sum of copies of a noncommutative ring (not only does it collide with the terminology for the direct sum of rings, it is not clear whether a left- or right module is meant); one should refer to the "direct sum of n copies of a ring as left modules". I believe that the left and right module structures that one might wish to attribute to a cartesian product are incompatible. I would argue that a module structure on the cartesian product of multiple copies of an arbitrary ring is unnatural (i.e. ill-defined), whereas the direct sum of n copies of a ring as a left module is well-defined. Authors like Bourbaki tend to be very careful with their wording in this regard. —Quondum 00:20, 20 May 2017 (UTC)

It is possible that I missed some subtleties but I didn't think the natural module structure there is tricky. I agree that we should be specific about whether R is viewed as a left or right module (basically the issue of Rⁿ when R is not commutative) but after that, the module structure seeems clear; first, it is an abelian group by component-wise addition. The scalar multiplication is $r\cdot (r_{e}|e\in E)=(rr_{e}|e\in E)$ (if R is a left module). The key point, as I understand, a direct sum of free modules is free while a cartesian product of them might not be; that's why we use a direct sum. -- Taku (talk) 03:00, 20 May 2017 (UTC)

The root of this seems to be the use of "cartesian product" to mean the direct product. Since cartesian product has a clear definition in WP (that does not imply structure beyond what is implied by the set), I would prefer the use of the term "direct product" when this is meant. Being someone who has first-hand experienced the amount of energy it takes to figure out when a term is used in a way that is not the defined meaning in WP (often requiring many-layered disambiguation), I guess I feel this more strongly than most. I see that you have at least linked the correct meaning, but this has a quality of hidden meaning that seems to be unnecessary. Some authors use terms loosely in this way (essentially forcing the reader to disambiguate through inference from context), but I think there is no reason to repeat such laziness here. —Quondum 15:35, 20 May 2017 (UTC)

I do not understand well the object of this discussion. It seems that it is about the respective uses of "direct product", "Cartesian product" and "direct sum". As far as know, many algebraists consider "direct product" and "Cartesian product" as synonymous, for the following reason. Normally, "direct product" means product in category theory. It is an easy theorem that for all algebraic structures which are varieties of universal algebra (that is all axioms are identities), the direct product is the Cartesian product of the underlying sets, equipped with the component-wise operations. This is the case for groups, modules, rings, ..., but not for fields. This theorem should appear, but is lacking, in Direct product.

On the other hand, for modules, a direct sum is the same thing as a coproduct, but for rings, a coproduct is a tensor product. Thus, if R is a ring, $R^{n}$ is, as a ring or left or right R-module, ..., the direct product of n copies of R. But, as a ring, $R^{n}$ is not a direct sum of n copies of R. Thus one should not talk of $R^{n}$ as a direct sum without clarifying that one considers only the module structure. In particular a free module has a natural ring structure only if and only its dimension is finite.

I hope that these remarks could be helpful for continuing to improve this article and other related articles. D.Lazard (talk) 18:00, 20 May 2017 (UTC)

This is a nuance of the way mathematicians use language (which, IMO, is shoddily, in the sense that adherent structure is often assumed while refusing to specify which structure is meant). What you have said here ("the direct product is the Cartesian product of the underlying sets, equipped with the component-wise operations") only implies that a direct product is a subcategory of a Cartesian product; this is like saying that a ring and a group are synonymous because a ring is a group equipped with a multiplication operation. Consideration of the automorphism group of an object should make this clearer. —Quondum 22:23, 20 May 2017 (UTC)