Talk:Jacobson radical

Untitled

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)

Two quotes from the text:

- The Jacobson radical of any field is {0}. The Jacobson radical of the integers is {0}.

- Unless R is the trivial ring {0}, the Jacobson radical is always a proper ideal in R

{0} is not a proper ideal. A field is not the trivial ring. What is going on? Juryu 16:28, 25 February 2006 (UTC)

Proper ideal of ring R here is evidently being used to mean ideal not equal to R.--CSTAR 17:02, 25 February 2006 (UTC)

Rings without identity

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)

Some comments on the changes

Thanks for doing these changes - the article looks a lot better. I don't think so many references are needed, however. According to WP:BURDEN, "All quotations and any material challenged or likely to be challenged" need an inline citation. Simple factual statements in articles such as this don't need them - a cursory scan doesn't reveal any controversial material. I think the better option would be to reference Isaacs' book in a References section at the end - happy for more experienced editors to correct me, however. I also removed the 'precisely' from the first sentence, as it seemed redundant.

I'm worried about it being written too much like a textbook in some places, such as "These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical." I'm not sure what would be better here. --Joth (talk) 20:48, 7 July 2009 (UTC)

Thankyou very much for your input. With regards to inline citations, I think you are correct. I will attempt to trim down some of the citations. It is best, in my view, if we reference more than one source. Once this is done, there is no need for repeating citations from the same author. On the other hand, I think that some inline citations are necessary, especially for the definition and various characterizations; although fewer are necessary that the current number.

I do not think that the article resembles a textbook in any particular way, although I could be wrong. The quote which you picked out is from the "intuitive discussion" section. When writing this article, I am trying to follow WP:MTAA to some extent, so that people who know what a ring is may understand this concept. I thought that it would be best to include an "intuitive" section so that such people could benefit. However, that section is not perfect, as well as being imprecise at times. I will try to rewrite it soon.

In general, I have expanded this article with the model, locally connected space, in mind. I think that that article has quite a few references also, but certainly not as many as this one. --PST 00:45, 8 July 2009 (UTC)

There is clearly a glut of references. Not every single sentence of the article needs to be referenced. One that stands out as particularly bizarre is the citation for the statement that J(R) will denote the Jacobson radical in this article. However, I do generally favor having references for specific statements of fact, like a proposition or theorem. Given that most of the article is sourced to just a few pages out of the Isaacs, it might be better to source entire paragraphs or sections rather than individual sentences anyway. Sławomir Biały (talk) 16:01, 9 July 2009 (UTC)

Too much like a text, not enough like an encyclopedia article

I want to point out that -- as with many math articles on wikipedia -- this one currently works well from an instructional perspective but not from an encyclopedic perspective. I am currently writing course notes on non-commutative algebra, and I wanted a reference to the original work of Jacobson in which the Jacobson radical is introduced. Not only does the current article not give any indication of this, it does not include any primary source material whatsoever. This is a problem...In fact, I have just looked back at the article and cannot find any link, reference or hint that the Jacobson radical is named after Nathan Jacobson! Plclark (talk) 20:33, 20 June 2011 (UTC)

Since this edit last year the article has stated that this radical is named after Nathan Jacobson. It says so in the last paragraph of the introduction. I do agree with your comment on the lack of historical references, though. Ozob (talk) 00:02, 21 June 2011 (UTC)

Thanks for pointing out the reference to Nathan Jacobson, which I had missed. Perhaps I missed it because it is several paragraphs down in the lede and appears in a paragraph which is concerned with the Jacobson radical for non-unital rings! Note also that the fact that "Jacobson was the first to study it" is a conspicuously unsourced assertion in an article that is otherwise meticulous to the point of fastidious with its sourcing. As a quick fix I moved this sentence to the end of the first paragraph. There is still a pointed lack of history and primary sources here. For what it's worth, I believe Jacobson's original paper is:

N. Jacobson, \emph{The radical and semi-simplicity for arbitrary rings.} Amer. J. Math. 67 (1945), 300--320.

Perhaps someone who is good with such things could add this to the article. Plclark (talk) 16:38, 21 June 2011 (UTC)

I added the technical parts in a believable way using the smallest changes I could. Feel free to organize them differently. JackSchmidt (talk) 17:40, 21 June 2011 (UTC)

Bourbaki reference

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)

Reorganizing the article

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)

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)

I might take a run at it in the next few days. Rschwieb (talk) 21:50, 28 June 2011 (UTC)

OK, started making big edits. For reference I am preserving an original copy at User:Rschwieb/JacobsonRadicalJune30-2011. Some large sections will be merely commented out. I will do my best to maintain all the material and to keep everything fully referenced. Rschwieb (talk) 16:24, 30 June 2011 (UTC)

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)

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)

Terminology in "intuitive explanation"

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)

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)

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)

(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)

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)

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)

Assessment comment

The comment(s) below were originally left at Talk:Jacobson radical/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The lead is now somewhat acceptable. We should now aim to illustrate the applications of this concept in module theory and non-commutative ring theory. More discussion is necessary on various characterizations of the Jacobson radical (how and why each characterization is useful in certain contexts, for instance). --PST 01:21, 7 July 2009 (UTC)

Last edited at 01:21, 7 July 2009 (UTC). Substituted at 02:15, 5 May 2016 (UTC)

I don't understand the first paragraph of "Geometric applications".

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)

Nakayama's lemma#Geometric interpretation has a probably more understandable explanation of this. -- Taku (talk) 11:24, 31 March 2022 (UTC)