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

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

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

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

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