Proposition 7

Let A be a semisimple ring.

a) Every finitely generated A-module is reflexive (II, §2, No. 7, p. 239).

b) For every simple left A-module S, the right A-module $S^*$ dual to S is simple, and the mapping $\lambda \mapsto$ cl($\lambda^*$) defines a bijection from the set of classes of simple A-modules to the set of classes of simple right A-modules.

c) Let M be a finitely generated left A-module. The right A-module $M^*$ dual to M is finitely generated and has the same length as M. Moreover, we have $[M : S] = [M^*: S^*]$ for every simple A-module S.

Let M be a finitely generated A-module.

By Proposition 4 of VIII, p. 138, the A-module M is finitely generated and projective; it is therefore reflexive by Corollary 4 of II, §2, No. 7, p. 240. In particular, every simple A-module is reflexive. It also follows that two finitely generated left modules are isomorphic if and only if their duals are.

Let S be a simple left A-module. Let $(T_i)_{i\in I}$ be a family of simple right A-modules with direct sum the dual $S^*$ of S. Since S is reflexive, it is isomorphic to $(S^*)^*$ and therefore to $\prod_{i\in I}T^*_i$. Each of the modules $T_i$ is reflexive; in particular, we have $T^*_i\not= 0$ for every $i\in I$. Since the simple module S is isomorphic to $\prod_{i\in I}T^*_i$, the set I has a single element, so $S^*$ is simple.

Since M is semisimple and finitely generated, it is the direct sum of simple submodules $S_1, . . . ,S_r$. Then $M^*$ is isomorphic to the direct sum of the family $S^*_1, . . . ,S^*_r$, and we have just seen that the modules $S^*_i$ are simple. Assertion c) follows immediately.