Proposition 1

Let A be a semisimple ring. There exists a finite family $(\mathfrak{m}_i)_{i\in I}$ of minimal left ideals of A such that $A_s=\oplus_{i\in I}\mathfrak{m}_i$. If $(\mathfrak{m}_i)_{i\in I}$ is such a family, then every simple A-module is isomorphic to one of the $\mathfrak{m}_i$. The set of classes of simple A-modules is finite.

The first assertion follows from the fact that the A-module $A_s$ is semisimple and finitely generated. Every simple module is isomorphic to a quotient of $A_s$ (VIII, p. 46, Proposition 1). The second assertion then follows from Corollary 3 of VIII, p. 56, and the third assertion is an immediate consequence of the second.