Proposition 8

Let A be a K-algebra, M an A-module, and L an extension of the field K. Denote the commutant of M by D and the center of D by Z.

a) Suppose that the $A_{(L)}$-module $M_{(L)}$ is simple (resp. isotypical, semisimple). Then the A-module M is simple (resp. isotypical, semisimple).

b) Suppose that the A-module M is semisimple and that M or L is finite-dimensional over K. The $A_{(L)}$-module $M_{(L)}$ is semisimple if and only if the ring $Z_{(L)}$ is reduced. The $A_{(L)}$-module $M_{(L)}$ is isotypical and nonzero if and only if the ring $Z_{(L)}$ is an integral domain.

c) Suppose that the A-module M is simple. The $A_{(L)}$-module $M_{(L)}$ is semisimple (resp. isotypical, simple) if and only if the ring $D_{(L)}$ is semisimple (resp. simple, a field).

Assertion a) is a specific case of Proposition 1 (VIII, p. 211), assertion b) is a specific case of Proposition 5 (VIII, p. 219), and assertion c) is a specific case of Corollary 1 of VIII, p. 215.