Corollary 2

Let L be an extension of the field K. Denote the center of the K-algebra A by Z.

a) If the L-algebra $A_{(L)}$ is semisimple, then the K-algebra A is semisimple.

b) Suppose that the K-algebra A is semisimple and that L or A is finite-dimensional over K. The L-algebra $A_{(L)}$ is semisimple if and only if the ring $Z_{(L)}$ is reduced; this is, in particular, the case when L is a separable extension of K. The L-algebra $A_{(L)}$ is simple if and only if the ring $Z_{(L)}$ is an integral domain; this is, in particular, the case when the center of A is equal to K.

Assertions a) and b) follow from Proposition 8, a) and b) applied to the A-module $A_s$.