The Stacks project

Definition 10.40.3. Let $R$ be a ring. Let $M$ be an $R$-module.

  1. Given an element $m \in M$ the annihilator of $m$ is the ideal

    \[ \text{Ann}_ R(m) = \text{Ann}(m) = \{ f \in R \mid fm = 0\} . \]

  2. 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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View Definition 10.40.3 as pdf