Definition 10.39.3. Let $R$ be a ring. Let $M$ be an $R$-module.

Given an element $m \in M$ the

*annihilator of $m$*is the ideal\[ \text{Ann}_ R(m) = \text{Ann}(m) = \{ f \in R \mid fm = 0\} . \]The

*annihilator of $M$*is the ideal\[ \text{Ann}_ R(M) = \text{Ann}(M) = \{ f \in R \mid fm = 0\ \forall m \in M\} . \]

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