About Module Homomorphism

Question

Let $R$ be a ring and $k$ be a subring of the center of ring $R$.

How to show that $\operatorname{Hom}_R(M,N)$ is a $k$-module in a canonical way?

Answer 1

If $f\in\operatorname{Hom}_R(M,N)$ and $a\in k$, define $$ af\colon M\to N,\qquad af\colon x\mapsto f(ax)=af(x) $$ It's just a matter of standard verifications to show that $af\in\operatorname{Hom}_R(M,N)$ and that this group has, in this way, the structure of $k$-module.

Be sure to point out where the property that $k$ is a subset of the center is used.