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semisimple algebra

Contents

Definition

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

Properties

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 (89.204.130.129)