Find a commutative ring R, a multiplicative subset S \subset R, and finitely generated R-modules M and N such that the natural map S^{-1}Hom(M,N) \to Hom(S^{-1}M,S^{-1}N) is not an isomorphism.
Funny, I was about to comment on how this FLMPotD is the first one in a while I feel well-equipped to solve. I didn't actually solve it in the first few minutes I looked at it (I tried coming up with some "simplest possible" example and then my modules ended up not being finitely generated), but maybe I will if I get a bit of spare time in the next couple of days.
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