Lemma 4

Let M be a left A-module with finitely generated countermodule. There exist a natural number $m$ and an injective $A_M$-linear mapping from $(A_M)_s$ to $M^m$.

Set E = End$_A(M)$. Let $(x_1, . . . , x_m)$ be a finite generating family of the E-module M. The mapping $\varphi :a\mapsto (ax_1, . . . , ax_m)$ from $(A_M)_s$ to $M^m$ is $A_M$-linear. Let $a$ be an element of $A_M$ such that $\varphi (a) = 0$. The set of elements $x$ of M such that $ax= 0$ is an E-submodule of M containing $x_1, . . . , x_m$ and is therefore equal to M, which implies $a= 0$.