Talk:Semisimple algebra

Algebra associative?
I assume that here in this context an algebra is meant to be associative, right? (AFAIK ideals in non-associative algebras don't make much sense.) Since it's not universally agreed upon whether an algebra should be associative or non-associative (e.g. the article Algebra (ring theory) takes all algebras to be associative while the article Algebra over a field assume them to be non-associative by default), I think it would be wise to explicitly say so. (ezander) 134.169.77.186 14:56, 7 May 2007 (UTC)

Finite dimensional? Jacobson radical
From Dales "Banach algebras and Automatic continuity" I have the definition of "semisimple" for a algebra to be that the Jacobson radical is zero. Is this equivalent to definition here in the finite dimensional case, or do we have different things with the same nameA Geek Tragedy 20:54, 25 May 2007 (UTC)?

OK I should read more carefully. It IS the same! In which case can we have the characterisation using the radical as the defintion (in the general case) and the current definition as "Theorem: If A is finite dimensional this is the same as..."? The general version is needed in some functional analysis settings.A Geek Tragedy 20:59, 25 May 2007 (UTC)

In the infinite dimensional case, both the definition of the radical and of semi-simplicity are different ! —Preceding unsigned comment added by 84.101.197.139 (talk) 19:46, 18 June 2008 (UTC)

Merge to semiprimitive ring?
It seems this article duplicates other articles such as semiprimitive ring, semisimple module, Artin–Wedderburn theorem and idempotent element. Thus, I propose we do some form of merger. -- Taku (talk) 01:52, 25 April 2013 (UTC)


 * It's probably just me but I like the functional analytic undertone of the article; this kinda resembles the analytic proof of classification of finite dimensional C*/W*-algebras. Be nice if you can preserve some of the POV when merging. Mct mht (talk) 03:31, 25 April 2013 (UTC)


 * That's actually a good point. I can clearly see the "tone" of this article differs from semisimple module. It may be "POV" but important "POV" to keep. -- Taku (talk) 13:21, 1 February 2015 (UTC)

Artinian
Erm, there are two definitions of semisimplicity, one that says the Jacobson radical has to be zero and a second that also requries the ring to be artinian. In the finite dimensional case these are the same, but the Artin--Wederburn theorem at the end requires the assumption that the ring be artinian as well. It is much better to incorporate artinianess into the definition.

Example: C[x] has trivial Jac rad but isnt a product of matrix rings — Preceding unsigned comment added by 129.215.104.100 (talk) 15:34, 21 July 2013 (UTC)

Not just division rings but division algebras
The article says:


 * The Artin–Wedderburn theorem completely classifies semisimple algebras: they are isomorphic to a product $$ \prod M_{n_i}(D_i) $$ where the $$ n_i $$ are some integers, the $$ D_i $$ are division rings, and $$ M_{n_i}(D_i) $$ means the ring of $$ n_i \times n_i $$ matrices over $$ D_i$$. This product is unique up to permutation of the factors.

However, I feel sure that "division ring" here should be "division algebra", so I'm going to change it. We're talking about algebras over a fixed field throughout this article, after all, and in this case the Artin-Wedderburn theorem gives not just division rings $$ D_i $$ but division algebras over that field. I was beginning to doubt my sanity, but a look at Knapp's chapter on Wedderburn-Artin ring theory convinced me I'm right John Baez (talk) 15:09, 21 July 2014 (UTC)

Cover semi-simple rings and not only algebras
Since a semi-simple algebra over a ring A (not necessarily a field) is just an A-algebra whose underlying ring is semi-simple, it would make perfect sense to:
 * Cover semi-simple rings first
 * Add material specific to semi-simple algebras over fields after that
 * Rename the article "Semisimple ring" and make a redirect space for "Semisimple algebra"
 * Add a "main article" link from Semisimple module

In addition, the article should make the important link to representation theory (of finite groups, say) explicit. And finally, there is clearly a need to make a wider cleanup of the cluster of articles on simple and semi-simple rings, algebras and modules and the related key theorems. Currently different articles follow conflicting terminology, in particular when it comes to requiring or not that simple rings be Artinian (<=> semisimple). This is only likely to confuse people following links from one to another. Stca74 (talk) 11:42, 1 December 2016 (UTC)