Lemma 10.5.2 (07JX)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.5: Finite modules and finitely presented modules
Lemma 10.5.2 (cite)

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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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