Talk:Jacobson radical

message: Isn't this list somewhat redundant?  A left primitive ideal is the same thing as the annihilator of a simple left module.
Waltpohl 04:57, 24 Feb 2004 (UTC)
replies: text: Proper ideal of ring R here is evidently being used to mean ideal not equal to R.--CSTAR 17:02, 25 February 2006 (UTC)
thread_title: Untitled

message: Listed under 'Properties' is this statement: "Unless R is the trivial ring {0}, the Jacobson radical is always an ideal in R distinct from R".  The Jacobson radical of R may be equal to R in a ring without identity.  -- Heath 69.174.67.197 15:14, 11 October 2006 (UTC)
thread_title:  Rings without identity 

message: The current reference is useless: "Bourbaki. Éléments de Mathématique."  Looking that up in mathsci.net gets you 70+ hits, and even with the right book a page should be given.  I can't see a reference to it right now either, so it may just be best to delete it. Rschwieb (talk) 00:50, 26 June 2011 (UTC)
thread_title:  Bourbaki reference 

message: Can the topmost introductory section be shortened, and the more technical contents merged with the sections below? This alone may help to diminish the glut of 19 references in the intro. As it is now, I feel the beginning contains too much information.  My philosophy is that the opening section should be as brief and clear as possible, and that the details should be organized into the body of the article. I am not really familiar with the popular philsophy for mathematics wiki articles, so it would be nice to get some feedback here on the idea. Rschwieb (talk) 13:55, 28 June 2011 (UTC)
replies: text: I agree.  I think the current lede isn't well-focused on basic information that the reader needs and instead drifts too far into technical details.  All of the content in it is good, just misplaced.  Ozob (talk) 21:27, 28 June 2011 (UTC)
replies: text: I might take a run at it in the next few days. Rschwieb (talk) 21:50, 28 June 2011 (UTC)

text: It looks much better already: good work!  Can you or someone else focus some attention on the "intuition" section?  A lot of it seems to be one person's personal thought processes about the Jacobson radical, which does not seem at all appropriate for an encyclopedia article.  There's a place for intuition: I am currently teaching a short graduate course on non-commutative algebra, and in class I said that the Jacobson radical is a "black hole around zero: if you're in the Jacobson radical you can't escape: multiplying by any element of the ring you are still so close to zero that when you add one you're so close to one that you're invertible".  But I did not put this in my lecture notes... Plclark (talk) 20:10, 30 June 2011 (UTC)
replies: text: Thank you!  I will defer altering/removing the intuition section to more experienced teachers and editors.  I find the most useful intuition to be about radicals being "bad" elements, but that of course is not unique to the Jacobson radical. As for the rest, I'm not sure it's really that helpful.  Maybe one possible workaround could be to copy it to an offsite page and link it in the references.  One good thing about this is that other mathematicians could contribute their own intuitions as well, if they can contact the site owner. Rschwieb (talk) 21:06, 30 June 2011 (UTC)
thread_title:  Reorganizing the article 

message: It is not clear what is a "module internal to the ring" or "act as a unit". Also, the connection alluded to between nilradical and Jacobson radical is very unclear and mostly just adds confusion.  As I have no idea what it should mean, I can't quite fix it. 94.112.136.34 (talk) 17:08, 15 April 2014 (UTC)
replies: text: This section should probably be renamed, and edited.  What is "intuitive" is highly subjective.  Also, the notion of "bad" element is highly POV.  --345Kai (talk) 20:35, 12 April 2016 (UTC)
replies: text: Does anyone mind if I remove this section or strongly edit it? I think the "bad" terminology is confusing and misdirects the reader. Furthermore, it seems to be motivated mainly by nilpotents, which is only part of the story for what Jacobson radicals "are" Wundzer (talk) 21:31, 24 December 2020 (UTC)
replies: text: (Late reply) The problem here is that it seems unreferenced. It is nice if we can motivate the notion, but that motivation has to be sourced by well-established texts. -- Taku (talk) 00:58, 8 January 2021 (UTC)
replies: text: Hey Taku, thanks for the input. I need to go through Hartshorne + Eisenbud for the correct references, but https://web.ma.utexas.edu/users/a.debray/lecture_notes/m392c_Raskin_AG_notes.pdf has a bunch of good applications of Nakayama's lemma (these should go into that article). I wonder if either of those references mention these relationships with the Jacobson radical. Otherwise, do you think it's fine to just cite Nakayama's lemma, and state how the Jacobson radical relates there? Wundzer (talk) 18:07, 8 January 2021 (UTC)
replies: text: No, I don't think the applications of Nakayama's lemma should be in this article. Nakayama's lemma is usually used for a local commutative ring and in that case the Jacobson radical is just the unique maximal ideal. The applications in the case of a local ring should thus be in the article local ring (or the Nakayama lemma article itself). The focus of this article should be the non-commutative case. -- Taku (talk) 01:40, 10 January 2021 (UTC)
thread_title:  Terminology in "intuitive explanation" 

message: Substituted at 02:15, 5 May 2016 (UTC)
thread_title: Assessment comment

message: Specifically, I don't understand how is the fact that a basis of a fiber at a point of a vector bundle extends to a basis of sections over a neighborhood of that point related in any way to Nakayama lemma or to the Jacobson radical. Mamuka Jibladze (talk) 12:57, 26 March 2022 (UTC)
replies: text: Nakayama's lemma#Geometric interpretation has a probably more understandable explanation of this. -- Taku (talk) 11:24, 31 March 2022 (UTC)
thread_title:  I don't understand the first paragraph of "Geometric applications".