Talk:Nakayama's lemma

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Matsumura (Commutative Ring Theory, Cambridge, 1996) mentions that Nakayama believed that this lemma be attributed to Azumaya and Krull; Matsumura uses NAK to denote this lemma. I think this should be mentioned. Kummini 20:59, 13 October 2006 (UTC)

Does a proof of the remark on coherent sheaves require NAK to prove? Take the case of an affine scheme $$ X = \mathrm{Spec} R$$ for some commutative ring $$R$$, $$F = \tilde{M}$$ for some finitely generated $$R$$-module $$M$$ and $$x$$ corresponds to a prime ideal $$ \mathfrak p \in \mathrm{Spec} R$$. Then the remark can be translated into the following: $$M_\mathfrak p = 0$$ if and only if there exists an ideal $$I \not \subseteq \mathfrak p$$ such that $$\forall \mathfrak q \not \supseteq I, M_\mathfrak q = 0$$. One direction is clear. Conversely, if $$M_\mathfrak p = 0$$, then there exists $$ f \not \in \mathfrak p, fM = 0$$ (since $$M$$ is finitely generated). Now set $$ I = (f) $$. Where did we use NAK? Kummini 21:22, 13 October 2006 (UTC)

The hypothesis is not that $$M_p=0$$ but that $$M_p/pM_p=0$$, which only implies $$M_p=0$$ by Nakayama. --128.252.164.145 (talk) 16:17, 30 April 2008 (UTC)

Who was Nakayama ?
The article doesn't state who the lemma was named after or rather has no details about him apart from his name. DFH 20:45, 27 January 2007 (UTC)

proof
it says that it's a corrolary of Cramer's rule, but i'm not sure that would be a short proof. anyways, it is a direct corrolary of the generalized version of Cayley-Hamilton theorem for modules. choosing fi=(identity of M), the characteristic polinomial coefficients are: 1, P1, ... Pn, where Pi in I. then 1+P1+...+Pn = 1 (mod I). --itaj 15:15, 11 March 2007 (UTC)

Major expansion of article
I have recently expanded the article. There is still more to be done, however. Possibly the most disputable aspect of the changes, is the fact that I have taken a view of Nakayama's lemma, from the perspective of the theory of Jacobson radicals. -- PS T  03:47, 9 July 2009 (UTC)


 * Nakayama's lemma nearly always refers to the version in commutative algebra, and so I think that the article should place due weight on this case, presenting the generalization later on in a generalizations section. Nakayama himself formulated this in the commutative case only, so it is at best misleading to call this version Nakayama's lemma.  Also, very little is actually gained by this generalization, whereas one must now be careful about distinguishing right from left modules.   And now the geometry of the lemma is lost (which is "really" about coherent sheaves... it has "nothing to do with" rings...)  So, while an expansion is warranted, it should be in the right direction from sources that reflect the most common uses of the lemma.  In addition to the sources already listed in the article, see, for instance, the book by David Eisenbud "Commutative algebra with a view towards algebraic geometry", or the definitive treatment by Bourbaki.  Sławomir Biały (talk) 04:35, 9 July 2009 (UTC)
 * I have commented at WikiProject mathematics. I will make a more specific comment here. First and foremost, let me stress that this article has not been significantly modified for about an year (there have been minor modifications, but mostly by robots). Therefore, at least my edits constitute some advancement in the quality of this article.
 * On the other hand, I do not claim that my version is better than the previous. With regards to the order of the article, I think that this is easy to change. I fully understand that Nakayama's theorem is one in commutative algebra (and geometry), but its version in non-commutative ring theory does have applications. For instance, one may prove that if S is a unitary subring of R that is central in R, and if R is finitely generated as a (natural) S-module, J(S) is a subset of J(R). The Jacobson radical is mainly a concept of non-commutative ring theory, for it is often not so interesting in the commutative case (for fields, of course (and more generally von Neumann regular rings), and therefore quite often in geometry, it is the zero ideal). Thus the non-commutative version does have its uses.
 * Since I have added some information of the non-commutative version of this lemma, I have improved the article, if anything. If you feel that the commutative case is neglected, please feel free to add some information on it, for I would be more than happy. However, since no one has few people have taken any interest in the article, criticizing improvements is perhaps less productive than improving the article yourself. -- PS T  05:36, 9 July 2009 (UTC)
 * Very well. (By the way, I did not come to this article because of the thread at WP:WPM, so clearly "some interest" did exist.) Sławomir Biały (talk) 05:46, 9 July 2009 (UTC)
 * The discussion is probably better kept here, unless you need to get third opinions. One point on the article: what is the actual statement of the lemma? I can see three Corollary but no lemma. --Salix (talk): 07:05, 9 July 2009 (UTC)
 * Let me thank Sławomir Biały for the improvements. I think that the article is far better now. -- PS T  01:06, 10 July 2009 (UTC)

Please look at the primary sources
In this article (as in many other wikipedia math articles), it seems that too little attention has been paid to the original sources. Nakayama's original article is

A remark on finitely generated modules Tadasi Nakayama Nagoya Math. J. Volume 3 (1951), 139-140.

It is available here: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.nmj/1118799226&page=record

He refers back to the article

On maximally central algebras

Gorô Azumaya

Source: Nagoya Math. J. Volume 2 (1951), 119-150.

It is available here: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.nmj/1118764746

In turn, Azumaya makes reference to

* The Radical and Semi-Simplicity for Arbitrary Rings * N. Jacobson * American Journal of Mathematics, Vol. 67, No. 2 (Apr., 1945), pp. 300-320

Jacobson's article is available on JSTOR.

Here are some things one learns by reading these articles that seem to be misrepresented in the wikipedia article:

1) Neither article uses a Cayley-Hamilton type argument for the proof, but rather a more direct inductive argument.

2) Both work in the context of noncommutative rings. (Indeed, Azumaya's article is one of the most influential articles ever written on the subject of noncommutative algebra.)  Therefore, presenting it primarily as a fact about commutative rings and claiming that it "partially generalizes" to the noncommutative case seems to misrepresent the history.

3) Nakayama's article is a short note which pushes around several equivalent statements. He in fact works in the context of a noncommutative ring which does not necessarily have a multiplicative identity, and much of the article is devoted to reductions from the case of an identity-less ring to a ring with identity.  In the case of a ring with identity, many of the statements are more obviously equivalent.  Statement II is what most sources nowadays call "Nakayama's Lemma".  This is the version which is stated (in contrapositive form) in the noncommutative section of the present article.  The statement and proof are the same as Theorem 1 in Azumaya's article.

4) Jacobson's article contains a result which is essentially the same (Theorem 10) but stated in terms of ideals.

I believe it would improve the article to take these points into account and to list the above papers as references. Especially, with regard to 3), I do not think that what is currently called "Nakayama's Lemma" is the right result to give that name. I do not know of any other authoritative reference that even includes the "there exists $$r \in R$$..." statement as part of Nakayama's Lemma.  Even Matsumura's book goes on to give the standard version, i.e., Statement II in Nakayama's paper for commutative unital rings.

If we are going to attribute the result to Krull, it would also be nice to track down that reference. Plclark (talk) 03:14, 9 August 2009 (UTC)
 * Thankyou for noting this. I had never thought to look at the original paper by Nakayama (but I will have a look now, since you have mentioned it). The old version of the article (prior to the edits by Slawomir Bialy) gave equal importance to both versions, but since Slawomir Bialy insisted that the commutative case was "more important", and I had no references to prove otherwise, the agreement was that the commutative case be given more weight. However, if the original paper gives equal weight to both cases, we should change the style of the article to match that of the original paper, rather than books on the subject. -- PS T  04:52, 9 August 2009 (UTC)


 * Typically the role of a Wikipedia article is to present material in proportion to its weight in secondary sources, not primary sources. Although it is interesting and helpful to have now some of the primary sources, I don't see that it necessitates any substantial changes to the article structure.  Sławomir Biały (talk) 12:27, 17 August 2009 (UTC)
 * I agree with your reasoning but I guess that my main criticizm was that the article reflected a view that Nakayama's lemma was two "separate lemmas"; one in the commutative case, and another in the noncommutative case. After all, any theorem in noncommutative ring theory also holds in commutative ring theory, so it is not absolutely vital that the commutative case be given more importance. Let me point out, however, that the "noncommutative version" of Nakayama's lemma does apply in geometry; specifically noncommutative algebraic geometry. Although I guess I am perhaps more inclined to geometric (topological) aspects of mathematics rather than algebraic, I still feel that it is incorrect to write all mathematics articles as if they are "for geometry" (for instance, an article on homotopy theory should not restrict to the case of manifolds only (but should instead define homotopy groups in the most abstract setting as possible), even though the algebraic topology of manifolds is the most well-known aspect of general algebraic topology). This is because it gives the incorrect impression that the work done by non-geometers is not as important as that done by geometers. Currently, this is being done too much in WP. -- PS T  03:56, 18 August 2009 (UTC)

Problematic statement about generalization
The Statement section says at the end “This conclusion of the last corollary holds without assuming in advance...” but this sentence is out of place, since it refers to the ideal ’’I’’ which does not appear in the last corollary. AxelBoldt (talk) 17:48, 13 June 2018 (UTC)
 * OK, I figured it out: someone rewrote the statements in terms of the Jacobson radical but did not at the same time adapt this sentence. I fixed it. AxelBoldt (talk) 18:05, 13 June 2018 (UTC)