Definition 3

Let A be a ring, M an A-module, and E = End$_A(M)$ the endomorphism ring of M. The countermodule of M is the left E-module with the same underlying additive group as M and external law $(c, x)\mapsto c(x)$.

Let Z be the center of the ring A. For every $a\in Z$, the homothety $a_M$ belongs to E. Consequently, E is canonically endowed with the structure of a Z-algebra. In particular, if M is a finitely generated Z-module, then the countermodule of M is finitely generated.