The Stacks project

Lemma 15.70.3. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. The following conditions are equivalent

  1. for every $a \in I$ the map $a : M \to M$ factors through a projective $R$-module,

  2. for every $a \in I$ the map $a : M \to M$ factors through a free $R$-module, and

  3. $\mathop{\mathrm{Ext}}\nolimits ^1_ R(M, N)$ is annihilated by $I$ for every $R$-module $N$.

Proof. The equivalence of (1) and (2) follows from Lemma 15.70.1. If (1) holds, then (3) holds because $\mathop{\mathrm{Ext}}\nolimits ^1_ R(P, N)$ for any $N$ and any projective module $P$. Conversely, assume (3) holds. Choose a short exact sequence $0 \to N \to P \to M \to 0$ with $P$ projective (or even free). By assumption the corresponding element of $\mathop{\mathrm{Ext}}\nolimits ^1_ R(M, N)$ is annihilated by $I$. Hence for every $a \in I$ the map $a : M \to M$ can be factored through the surjection $P \to M$ and we conclude (1) holds. $\square$

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