Talk:Semisimple algebra

message: 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)
thread_title:  Algebra associative? 

message: 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)
thread_title: Finite dimensional? Jacobson radical

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Merge to [[semiprimitive ring]]? 

message: 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)
thread_title:  Artinian 

message: 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)
thread_title:  Not just division rings but division algebras 

message: 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)
thread_title: Cover semi-simple rings and not only algebras