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.

Last revised on October 14, 2012 at 16:50:26.
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