Proposition 5

Let A be a left Artinian ring and M a faithful A-module.

a) There exist a natural number $m$ and an isomorphism from $A_s$ to a submodule of $M^m$.

b) If M admits a Jordan–Hölder series $(M_i)_{0\leqslant i\leqslant n}$, then every simple A-module is isomorphic to one of the quotients $M_i/M_{i+1}$.

By VIII, p. 2 applied to the A-module $A_s$ and the annihilators of the elements of M, there exist elements $x_1, . . . , x_m$ of M such that the annihilator of M is the intersection of the annihilators of the $x_i$. Since the A-module M is faithful, the A-linear mapping $a\mapsto (ax_1, . . . , ax_m)$ from $A_s$ to $M^m$ is injective, whence a).

Under the assumption of b), every simple quotient of $A_s$ is isomorphic to a quotient of a Jordan–Hölder series of $M^m$ (I, §4, No. 7, p. 43, Corollary), hence to one of the modules $M_i/M_{i+1}$ (I, §4, No. 7, p. 43, Theorem 6). We conclude that every simple A-module is isomorphic to a quotient of $A_s$.