Lemma 1

Let A be a simple ring, S a simple left A-module, and D the opposite ring of the field End$_A(S)$. Then S is an invertible $(A,D)$-bimodule. It is also a finite-dimensional right vector space over D, and the mapping $a\mapsto a_S$ is a ring isomorphism from A to End$_D(S)$.

By Proposition 1, the A-module $A_s$ has finite length and is isotypical of type S. Hence, there exists an integer $m\geqslant 1$ such that the A-modules $A_s$ and $S^m$ are isomorphic. Then the A-module S is projective and finitely generated. It is generating (VIII, p. 80, Theorem 1), and Lemma 1 follows from Theorem 1 of VIII, p. 101, (ii)$\Rightarrow$(i) and (ii)$\Rightarrow$(iii) applied to the $(A,D)_{\mathbf{Z}}$-bimodule S.