Proposition 2

Let A be a simple ring and S a simple A-module.

a) Every A-module is projective and isotypical of type S, hence semisimple. If it has length $\mathfrak{a}$, then it is isomorphic to $S^{(\mathfrak{a})}$.

b) Every nonzero A-module is generating.

c) Two A-modules are isomorphic if and only if they have the same length.

Denote the opposite field of the commutant of S by D. By Lemma 1 of VIII, p. 120, S is an invertible $(A$, D)-bimodule. Let M be an A-module; by Lemma 3, it is isomorphic to a module of the form $S\otimes_DV$, where V is a left vector space over the field D.

The D-module V is projective and isotypical of type $D_s$; it is generating if and only if it is not reduced to 0. Finally, the length of $S\otimes_DV$ is equal to the dimension of the D-vector space V, and two vector spaces are isomorphic if and only if they have the same dimension. Proposition 2 now follows in view of Propositions 10 of VIII, p. 109 and 12 of VIII, p. 110.

Let $r\geqslant 1$ be an integer. We say that a cardinal $\mathfrak{a}$ is divisible by $r$ if there exists a cardinal $\mathfrak{b}$ such that $\mathfrak{a}=r\mathfrak{b}$. This is the case if $\mathfrak{a}$ is infinite because we have $r\mathfrak{a}=\mathfrak{a}($Set Theory, III, §6, No. 3, p. 188, Corollary 3). It follows from this remark that if the cardinal $\mathfrak{a}$ is divisible by $r$, then there exists a unique cardinal $\mathfrak{b}$ such that $\mathfrak{a}=r\mathfrak{b}$.