Proposition 2

Let K be a commutative field, A a K-algebra, and M an A-module that is finite-dimensional over K. The following properties are equivalent:

(i) The A-module M is absolutely semisimple.

(ii) There exists an extension P of K that is a perfect field such that the $A_{(P)}$-module $M_{(P)}$ is semisimple.

(iii) The A-module M is semisimple, and the center of its commutant is an étale algebra over the field K.

It is clear that (i) implies (ii). Assume that (ii) holds; then the $A_{(P)}$-module $M_{(P)}$ is absolutely semisimple by Corollary 1, a) of VIII, p. 222. By Proposition 1, b), M is then absolutely semisimple. Therefore, (ii) implies (i).

Suppose that M is semisimple. Let Z be the center of the commutant of M; it is a commutative algebra of finite degree over the field K. If L is an extension of K, then it follows from Proposition 8, b) of VIII, p. 222 that the $A_{(L)}$-module $M_{(L)}$ is semisimple if and only if the ring $Z_{(L)}$ is reduced. The equivalence of (i) and (iii) therefore follows from Theorem 4 of V, §6, No. 7, p. 34.