Where am I using $R$ is an integral domain in proof of: If $N$ is torsion-free, then $\mathrm{Hom}_R(M,N)$ is torsion-free.

Question

Let $R$ be an integral domain and let $M$ and $N$ be $R$-modules. If $N$ is torsion-free, then $\mathrm{Hom}_R(M,N)$ is torsion-free.

My proof is:

Let $f\in \mathrm{Hom}_R(M,N)$ be non-zero. Then $rf \in \mathrm{Hom}_R(M,N)$. If $rf=0$, then $rf(m)=(rf)(m)=0$ for all $m \in M$. Since $N$ is torsion-free, then $r=0$. So, $\mathrm{Hom}_R(M,N)$ is torsion-free.

I don't see where I am using the assumption that $R$ is an integral domain. Is my proof not finished?

Answer 1

"Torsion-free" for non integral domains is more subtle because if $rs = 0$ in $R$ then $rsm = 0$ for all $m \in M$. Either $sm = 0$ or $r(sm) = 0$. Because of this, we have to define a torsion-free module as a module where $rm = 0$ implies $r = 0$ only when $r$ is regular (not a zero-divisor).

Your proof works for non integral domains provided that $r$ is a regular element and we interpret "torsion-free" correctly.