Talk:Simple ring
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[edit]The comment(s) below were originally left at Talk:Simple ring/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 The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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| Probably start class. Consider removing stub tag and uprating. Geometry guy 16:27, 22 May 2007 (UTC) The characterization of minimal left ideals of the matrix ring M(n,D) in the current version is false. It is not true that every minimal left ideal is one of those presented. Those are examples, but they are not all the minimal left ideals. Given any elements r_1, r_2,... r_n, of the division ring, not all zero, a minimal left ideal arises, consisting of all nxn matrices having each row of the form x*r_1, x*R_2,... x*r_n, for various x depending on the rows. This is a left ideal, because action by the ring from the left only performs row operations. Every nonzero element of such an ideal generates the ideal, so these are minimal. They do not take the form given, unless all but one r_i is zero. I suggest changing the article to clearly state that the minimal left ideals exhibited are examples only, which nevertheless do motivate the sketched proof of Wedderburn's theorem that follows it. Drwesb (talk) 01:20, 20 July 2010 (UTC) | |
Last edited at 01:20, 20 July 2010 (UTC). Substituted at 02:36, 5 May 2016 (UTC)
Merge from simple ring
[edit]As far as I can tell, simple algebra and simple ring discuss the same concept; hence, the merger is in order. -- Taku (talk) 01:01, 10 January 2020 (UTC)
These two concepts are not the same. Most of the current version of simple algebra discusses the concept in universal algebra, of which both simple rings and simple algebras (in the sense of modules equipped with a bilinear multiplication) are special cases. caterpillar_tree (talk) 16:13, 28 April 2020 (UTC)
- Sorry, I meant to say except the section on universal algebra. The problem is that currently, despite the article title, the article simple ring is mostly about a simple algebra. I didn’t mean to say the two concepts are the same; I was referring to materials. I have gone ahead with the partial merger and renamed the simple algebra to simple universal algebra (since I doubt the primary meaning of a simple algebra is one in universal algebra). Simple algebra now redirects to simple ring. —- Taku (talk) 16:35, 28 April 2020 (UTC)
Ideals in matrix rings
[edit]It's not true that any left ideal of a matrix ring $M_n(D)$ has the claimed form. The left ideals correspond to $D$-linear subspaces of $D^n$, where a $D$-linear subspace $L \subset D^n$ is associated to the ideal of $M_n(D)$ where each row has entries from $L$. Not all such subspaces are given by setting a subcollection of coordinates to zero, even when $D$ is a field. — Preceding unsigned comment added by 2601:246:100:2B00:5430:9CE2:C024:D606 (talk) 18:35, 11 January 2022 (UTC)
- A simple counterexample to the claim is the ideal . This is clearly an ideal that doesn't fit the article's description. --Svennik (talk) 18:45, 7 February 2022 (UTC)
![{\displaystyle I=\{[v|v|\dotsb |v]:v\in D^{n}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a55a8a79301a107b0281a16ab2fdc070653a4c06)
- Hold on. It's incredible that this mistake has existed since circa 2005. Just look at the history of this article! Does anybody actually check this stuff? --Svennik (talk) 18:52, 7 February 2022 (UTC)
I haven’t checked the proof myself but I want to note that the proof that a simple Artin ring is a matrix ring over a division ring also appears in Artinian ring. So, I am assuming the proof here should also be ok, up to some corrections if needed. —- Taku (talk) 14:14, 22 March 2023 (UTC)
- The proof of Wedderburn-Artin on this page was completely bogus, for the reason Svennik explained. I deleted it and added a bunch of new material on simple rings. There is a proof of Wedderburn-Artin on the page Wedderburn-Artin theorem, and I think that's the right place for this proof. John Baez (talk) 19:15, 12 June 2023 (UTC)