Example 3

Let M be a semisimple A-module whose countermodule is finitely generated. Then M is a generating $A_M$-module. Indeed, by Lemma 4 of VIII, p. 8, there exists a natural number $m$ such that $(A_M)_s$ is isomorphic to a submodule of $M^m$. Since $M^m$ is a semisimple $A_M$-module, $(A_M)_s$ is isomorphic to a direct factor submodule of $M^m$ and M is a generating $A_M$-module (Theorem 1).