I'm stuck with the dual of an exercise in Atiyah-MacDonald.
It's easily seen that tensoring over $R$ with $-\otimes M$ the sequence $$ 0\to \mathfrak a\to R\to R/\mathfrak a\to 0 $$ one gets the isomorphism $R/\mathfrak a\otimes M\cong M/\mathfrak aM$. What can I deduce from "Hom(-,M)ing" the same sequence? I have, let's say $$ 0\to \mathfrak a\xrightarrow{\alpha} R\xrightarrow{\beta} R/\mathfrak a\to 0 $$ and $$ 0\to \hom(R/\mathfrak a,M) \xrightarrow{\beta^*} \hom(R,M)\cong M\xrightarrow{\alpha^*} \hom(\mathfrak a,M) $$ but now what can I do? What is $\hom(R/\mathfrak a,M)$ like?
I know, it's easy, but...