Example 4

Let A be a principal ideal domain, and let P be a finitely generated A-module. There exist an integer $n\geqslant 1$ and an increasing sequence of ideals $(\mathfrak{a}_i)_{1\leqslant i\leqslant n}$ of A such that P is isomorphic to the direct sum of the $A/\mathfrak{a}_i$ (VII, §4, No. 4, p. 19, Theorem 2); the annihilator $\mathfrak{a}$ of P is equal to $\mathfrak{a}_1$. Then P is a generating module over the ring $A/\mathfrak{a}$. If P is not a torsion module, then we have $\mathfrak{a}= 0$ and P is a generating A-module.