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...