semisimple algebra



An associative unital algebra AA over a field kk is semisimple if its Jacobson radical is trivial.


If AA is finite-dimensional, this is equivalent to saying that AA is a finite product of finite-dimensional simple algebras.

By the Artin-Wedderburn theorem?, any finite-dimensional simple algebra over kk is a matrix algebra with entries lying in some division algebra whose center is kk. So, every finite-dimensional semisimple algebra is a finite product of such matrix algebras.

Revised on October 14, 2012 16:50:26 by Urs Schreiber (