Example

Let A be a principal ideal domain, K its field of fractions. The A-module K/A is a Matlis module: in fact the canonical mapping from K/A into $\prod_{m \in \Omega} K/A_m$ induces an isomorphism from K/A onto $\bigoplus_{m \in \Omega} K/A_m$ (A, VII, p. 10, th. 2); the assertion then follows from No. 1, example 1. We have moreover already proved in A, VII, § 4, No. 9 that the mapping $\alpha_M$ is bijective for every $A$-module M of finite length when the ring A is principal.