Remark 4

An $A$-module is called simple ($I$, p. 37) if it is nonzero and has no submodules other than itself and $0$; it is then necessarily cyclic, so finitely generated, and indecomposable; since the modules $A/(p^n)$ are not simple for $n \neq 1$, while the modules $A/(p)$ are, and since $A$ is simple only if the ring $A$ is a field, we deduce that the simple $A$-modules are: a) free modules of rank 1, when $A$ is a field; b) modules isomorphic to quotients $A/(p)$, where $p$ is an irreducible element of $A$, when $A$ is not a field.