Talk:Nakayama's lemma

message: 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)
thread_title: [Untitled]

message: 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)
thread_title:  Who was Nakayama ? 

message: 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)
thread_title:  proof 

message: 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)
replies: text: 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. --PST 04:52, 9 August 2009 (UTC)
replies: text: 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)
replies: text: 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. --PST 03:56, 18 August 2009 (UTC)
thread_title:  Please look at the primary sources 

message: 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)
replies: text: 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)
thread_title:  Problematic statement about generalization