Separable algebra

Summary

A separable algebra over a commutative ring $K$ is an associative algebra $A$ whose multiplication map $A \otimes_K A^{\mathrm{op}} \to A$ splits as a map of bimodules. Over a field, for finite-dimensional $A$, it means that $A$ stays semisimple after every extension of the base field. It generalizes the notion of a separable field extension to noncommutative algebras, and it is the starting point of the theory of Azumaya algebras.