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).

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