Lemma 15.88.3. Let $R$ be a ring. Let $I$ be an ideal of $R$. For any $R$-module $M$ set $M[I^ n] = \{ m \in M \mid I^ nm = 0\} $. If $I$ is finitely generated then the following are equivalent

$M[I] = 0$,

$M[I^ n] = 0$ for all $n \geq 1$, and

if $I = (f_1, \ldots , f_ t)$, then the map $M \to \bigoplus M_{f_ i}$ is injective.

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