Proposition 4

Let K be a commutative field and A a K-algebra.

a) Let M be a semisimple A-module. The A-module M is absolutely semisimple if and only if every simple module belonging to the support of M is.

b) Let S be a simple A-module, and let D be its commutant. The following properties are equivalent:

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

(ii) The K-algebra D is absolutely semisimple.

(iii) The K-algebra D is a field, has finite degree over K, and its

center is a separable extension of K.

Assertion a) follows from Proposition 1, a) of VIII, p. 229. Let S and D be as in b), and let L be an extension of K. The $A_{(L)}$-module $S_{(L)}$ is semisimple if and only if the ring $D_{(L)}$ is (VIII, p. 222, Proposition 8, c)). This proves the equivalence of (i) and (ii), and that of (ii) and (iii) follows from Theorem 1 because D is a field.