Lemma 15.70.1. Let $R$ be a ring. Let $M$, $N$ be $R$-modules.

Given an $R$-module map $\varphi : M \to N$ the following are equivalent: (a) $\varphi $ factors through a projective $R$-module, and (b) $\varphi $ factors through a free $R$-module.

The set of $\varphi : M \to N$ satisfying the equivalent conditions of (1) is an $R$-submodule of $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$.

Given maps $\psi : M' \to M$ and $\xi : N \to N'$, if $\varphi : M \to N$ satisfies the equivalent conditions of (1), then $\xi \circ \varphi \circ \psi : M' \to N'$ does too.

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