Lemma 15.70.2. Let $R$ be a ring. Let $\varphi : M \to N$ be an $R$-module map. If $\varphi $ factors through a projective module and $M$ is a finite $R$-module, then $\varphi $ factors through a finite projective module.
Proof. By Lemma 15.70.1 we can factor $\varphi = \tau \circ \sigma $ where the target of $\sigma $ is $\bigoplus _{i \in I} R$ for some set $I$. Choose generators $x_1, \ldots , x_ n$ for $M$. Write $\sigma (x_ j) = (a_{ji})_{i \in I}$. For each $j$ only a finite number of $a_{ij}$ are nonzero. Hence $\sigma $ has image contained in a finite free $R$-module and we conclude. $\square$
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