## The homomorphic image of a torsion element is torsion

Let $R$ be a ring, let $M$ and $N$ be left $R$-modules, and let $\varphi : M \rightarrow N$ be an $R$-module homomorphism. Prove that $\varphi[\mathsf{Tor}_R(M)] \subseteq \mathsf{Tor}_R(N)$.

Let $m \in M$ be a torsion element, with $r \in R$ nonzero such that $r \cdot m = 0$. Now $r \cdot \varphi(m) = \varphi(r \cdot m)$ $= \varphi(0) = 0$, so that $\varphi(m) \in N$ is torsion. Thus $\varphi[\mathsf{Tor}_R(M)] \subseteq \mathsf{Tor}_R(N)$.