nLab semisimple algebra

Definition

An associative unital algebra $A$ over a field $k$ is semisimple if its Jacobson radical is trivial.

Properties

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

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