Talk:Finitely generated module/Archive 1

A modification of the previous example gives a finitely-generated module: take the rational numbers whose denominator is strictly less than 6 (after simplification: 16/12=4/3, so it belongs to this set). This is a module over the integers, which is also finitely generated. A set of generators is, for example, {1/1,1/2,1/3,1/4,1/5}, but also {1/3,1/4,1/5} (this one is minimal).

This isn't a module. Since 1/4 is cited as part of a minimal generating set, I assume the intention is that addition in the module is addition of rationals, and multiplication by a coefficient is ordinary multiplication.

In that case, observe that 1/4 + (-1)*(1/5) = 1/20 is not in the set described. So the set described isn't closed and hence is not a module.

Also, in "modifying the previous example" the meanings of addition and multiplication in the module have changed. This should be explicitly stated.

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SORRY!!!!!!! Sorry. Sorry. Sorry. I have been misguided for my constant thinking on semigroups and things with a common denominator. You are absolutely right. I am going to re-edit it right now. Pfortuny

What I wanted to do was showing a simple example of a f.g. module for which any set of generators is linearly dependent. This is trivial with torsion modules, but I did not want to mess up a casual reader. Any ideas? Pfortuny

Example 1

How is a map (of the Earth, I assume) supposed to be a module??? Over which ring? --129.70.14.127 23:49, 7 November 2007 (UTC)

The free group on two generators... Well, why is it then on two generators? Commentor (talk) 00:16, 22 March 2008 (UTC)

Rewrite the whole article!

The whole article is actually a real mess! I don't know how to fix it. The "intuitive introduction" gives three examples, all of which either make no sense or are plainly wrong! It seems, the author did not understand what a module is. For example, {2,3} does not generate 1/6 over the integers.

Moreover, there is no actual introduction there. Instead there is some plain nonsense about directions and distances that has nothing to do with finitely-generated modules. I would delete all of that. The really important examples are not given. The rest is given in a random, chaotic order. Definitions are mixed with lemmas and observations. The relation to Noetherian rings is not clearly stated. Basic facts are not highlighted. Commentor (talk) 00:38, 22 March 2008 (UTC)

Most of the article is fine. The lead paragraph needs to be written; currently there is only a short opening sentence. The intuitive introduction is a bit too verbose, and uses examples that may not be clear to some people, but the examples are correct. The article lacks sources, and not all of the claims have appropriate citations (many link to a wiki article precisely explaining the claim, so need no further support in this article). The section on coherent modules could use some expansion, and linking to coherent ring.

The examples with rational numbers are using the positive rational numbers as a concrete version of the direct sum of countably many copies of the integers. The underlying abelian group of the positive rationals is the one given by multiplication. It is a free abelian abelian group with basis the positive rational integer primes. The Z-module structure is given by exponentiation, so that the action of the scalar n on the module element (p/q) results in the module element (p/q)^n. It is not clear to me why the example did not use the nonzero rationals instead, as it is incredibly common to consider the multiplicative group of a number ring as a module over the integers, and compute its torsion part, as in Dirichlet's unit theorem.

You might try adding a more standard examples of finitely generated and non finitely generated modules in a plain old "examples" section. JackSchmidt (talk) 20:00, 22 March 2008 (UTC)

Thanks! Could you, please, also explain examples 1 and 3? Commentor (talk) 03:58, 23 March 2008 (UTC)

I believe example 1 is trying to indicate the free module of rank 2 over the ring of integers, that is, just Z^2 as a Z-module. The "integer lattice" picture is common. I think the informality of this example belongs more in an introductory article on abelian groups.

I believe example 3 is trying to indicate the submodule of the positive rationals (under multiplication and exponentiation) generated by 2 and 3.

Both the examples are poorly worded.

I would also prefer a few explicit examples based on change of ring. The rationals (under addition) are not finitely generated over Z, but they are over Q. The module R[X] of polynomials in one variable over a ring R is not finitely generated as an R-module, but is as an R[X] module.

Some modules are not finitely generated over any ring (they are not finitely generated over their endomorphism ring). A nice example is the abelian group under addition consisting of all rational numbers whose denominators (in lowest terms) are not divisible by the square of any prime. JackSchmidt (talk) 04:13, 23 March 2008 (UTC)

Now I understand. This is indeed too informal (reminds me this presentation: http://www.youtube.com/watch?v=nLcr-DWVEto) I was confused because... I thought a map is a sphere! It would be more reasonable to say that e.g. every ring is finitely generated over itself, and then "more genrally, the direct sum of a finitely many copies of a ring is f.g., thus $Z^{2}$ is generated by (1,0) and (0,1)". This should be enough. Also, your point about endomorphism ring is a good point (though I never thought/heard this idea before about f.g. over any ring vs the endomorphism ring) - should be worth including in the article. Commentor (talk) 00:23, 24 March 2008 (UTC)

Did some editing. I am not satisfied with "Some facts", as it is not logically structured and perhaps does not mention all facts that are worth mentioning. Also, the section on finitely presented and coherent modules is not good either. (Maybe it is worth to have one or two separate articles and then to give only a short account here). Overall, no general comments are present about the role of the concept of a finitely generated module in the general picture. Hopefully, someone can do all that. Commentor (talk) 00:33, 24 March 2008 (UTC)

The example fixes you made were just great: very little change to the text, yet now the examples are suddenly clear. I'll look at the rest of the article. I won't have time to expand the coherent module section for a while (there are some nice papers of Chase to have around when doing this, and there is an injective-flat duality that is particularly transparent for coherent rings). I think commutative algebraists should be able to say something interesting about integral closure and algebras that are finitely generated as modules. Finitely generated projective modules are important in Morita equivalence. JackSchmidt (talk) 14:30, 24 March 2008 (UTC)

Comment

I just want to give my quick impression. (I might try to edit later.) To me, the article reads more on the generalization of finitely generated modules. I think we need more basic materials like finite morphism, relation between "finite" and "of finite type". Integality. etc. Especially on coherent module (but I'm not an expert-enough to write on these.) -- Taku (talk) 14:33, 17 August 2011 (UTC)

Hi, I just wanted to ask a few things. Can you rephrase "reads more on the generalization of finitely generated modules" for me? (Maybe 'needs more'?) Module theoretically, the article contains almost all the basics. Is 'finite morphism' basic fare for algebraic geometry? (I'm wouldn't know, I haven't ever encountered it.) If so, it would be nice to have a section explaining the connections. Lastly, Hilbert's Basis Theorem seems to be tucked away in "Some facts" under this guise: "However, if the ring R is Noetherian, then every submodule of a finitely generated module is again finitely generated (and indeed this property characterizes Noetherian rings)." Rschwieb (talk) 19:42, 17 August 2011 (UTC)

Yes, I missed Hilbert basis. (I will add a link). As to "basic facts", I meant to say the focus on the article seems on the concepts related to f-gen modules rather than f-gen modules themselves, like examples, modules over a graded ring. (Maybe not "basic" but "central".) It's simpler for me to just edit. (I hope I don't mess things too much.) -- Taku (talk) 12:43, 18 August 2011 (UTC)

OK thanks, that helps a lot :) Rschwieb (talk) 15:23, 18 August 2011 (UTC)

(Basically a note to myself). There is a theorem due to Artin-Tate that relates a f-gen module and f-gen algebra. Can't remember the exact statement. It's basically a variant of Hilbert nullstellensatz. -- Taku (talk) 13:31, 6 September 2011 (UTC)

Your note motivated me to add the Kaplansky content I just added, but I have no idea if that's what you were thinking of. Rschwieb (talk) 15:56, 6 September 2011 (UTC)

Example on fractional ideals

Currently the first example in this article is "Let R be an integral domain with K its field of fractions. Then every R-submodule of K is a fractional ideal. If R is Noetherian, every fractional ideal arises in this way." I am confused about the role of this example, since finitely generated modules are not mentioned (and fractional ideals need not be finitely generated, any more than usual ideals). I prefer not to remove it, since I'm probably missing something. If there's someone who understands what this is about, could you edit it to clarify the connection with finitely generated modules? Tesseran (talk) 18:08, 13 June 2014 (UTC)

No the problem is "finitely generated" was missing. I fixed it. -- Taku (talk) 19:29, 13 June 2014 (UTC)

Any module is a union of an increasing sequence of finitely generated submodules.

Consider the following R-module. The underlying group is the set of maps f:R->R of the reals into the reals with the property that for any element f, {x in R; f(x)!=0} is finite or empty. Addition is defined by

(f1+f2)(x)=f1(x)+f2(x), with identity element e(x)=0 for all x in R.

Finally, we define (y*f)(x)=y*f(x).

Perhaps I'm missing something, I don't see how this module could possibly be the union of an increasing sequence of finitely generated submodules.Kerry (talk) 23:59, 13 December 2014 (UTC)

I don't think a module is a union of a "sequence" of f-gen modules in general; but the article says a union of an increasing chain of f-gen modules. So, I think it's ok. In the above example, let $E_{i}$ be an increasing chain of finite subsets of M whose union is M. Let $M_{i}$ be the submodule of M generated by $E_{i}$. Then $M_{i}$ has the required property. -- Taku (talk) 17:43, 27 June 2015 (UTC)