Corollary 2

Let $A$ be a ring which is not necessarily commutative, $m$ a two-sided ideal of $A$ contained in the Jacobson radical of $A$, $M$ a left $A$-module and $(x_i)_{i \in I}$ a family of elements of $M$. If $M$ is finitely generated or $m$ is nilpotent and the elements $1 \otimes x_i$ ($i \in I$) generate the left $(A/m)$-module $(A/m) \otimes_A M$, the $x_i$ generate $M$.

Let $(e_i)_{i \in I}$ be the canonical basis of the left $A$-module $A_s^{(I)}$: it is sufficient to apply Corollary 1 to the $A$-linear mapping $u : A_s^{(I)} \to M$ such that $u(e_i) = x_i$ for all $i \in I$.