Let $R$ be a ring, and $M,N$ be left $R$-modules. Then is it not true that $\operatorname{Hom}_R(M,N)$ has the structure of an $R$-module?
I was reading the preface of the Homological Algebra book by Rotman and was quite surprised to learn that this is not the case. I think all the axioms for being a module are satisfied by $\operatorname{Hom}_R(M,N)$, but Rotman is very unlikely to make a mistake. What is it that I am missing?
Under what circumstances is this true?