I'm working on an assignment problem and I'm stuck.
Statement of Problem:
Let $R$ be a Noetherian ring, and $S\subset R$ a multiplicatively closed subset. Show that if $M$ and $N$ are $R$-modules, with one of them finitely generated, then show that
$S^{-1}Hom_{R}(M,N)$ is isomorphic to $Hom_{S^{-1}R}(S^{-1}M, S^{-1}N)$
The "natural isomorphism" that I tried to define (that is, take an element $\frac{f}{s}$ in $S^{-1}Hom_{R}(M,N)$ and map it to the function $\phi$ defined by $\phi(\frac{m}{s}) = \frac{f(m)}{s}$) is not injective ($\phi$ "forgets" about the $s$ i started with in the denominator).
I'm actually really stumped on how to show this. For starters, the hypothesis of one of $M$ and $N$ being finitely generated throws me off as I'm not sure how that will be involved. Additionally, I'm not even sure if I should be trying to show it directly even. On a previous assignment, I proved an isomorphism existed by constructing the mapping and showing it was an isomorphism. After two pages, the solution was two lines using an exact sequence. :P
Any advice?