Example 2

Let $a$ be a nonzero element of $A$. The map $x \mapsto x/a$ from $A$ into $K$ induces an isomorphism on quotient modules from $A/(a)$ onto the submodule $(K/A)(a)$ of $K/A$ consisting of elements annihilated by $a$. If $M$ is an $A$-module annihilated by $a$, or equivalently an $A/(a)$-module, then the $A$-module $D(M)$ is identified with $\operatorname{Hom}_{A/(a)}(M, A/(a))$. We leave the reader to rewrite the preceding results for this special case in the corresponding notation (cf. V, p. 86).