Lemma 10.5.2. Let $R$ be a ring. Let $\alpha : R^{\oplus n} \to M$ and $\beta : N \to M$ be module maps. If $\mathop{\mathrm{Im}}(\alpha ) \subset \mathop{\mathrm{Im}}(\beta )$, then there exists an $R$-module map $\gamma : R^{\oplus n} \to N$ such that $\alpha = \beta \circ \gamma$.

Proof. Let $e_ i = (0, \ldots , 0, 1, 0, \ldots , 0)$ be the $i$th basis vector of $R^{\oplus n}$. Let $x_ i \in N$ be an element with $\alpha (e_ i) = \beta (x_ i)$ which exists by assumption. Set $\gamma (a_1, \ldots , a_ n) = \sum a_ i x_ i$. By construction $\alpha = \beta \circ \gamma$. $\square$

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