Separable algebra

In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

Let K be a field. An associative K-algebra A is said to be separable if for every field extension ${\displaystyle \scriptstyle L/K}$ the algebra ${\displaystyle \scriptstyle A\otimes _{K}L}$ is semisimple.

There is a classification theorem for separable algebras: separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field K. In particular: Every separable algebra is itself finite-dimensional. If K is a perfect field --- for example a field of characteristic zero, or a finite field, or an algebraically closed field --- then every extension of K is separable. As a result, if K is a perfect field, separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras over field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K.

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