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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an $R$-[[module|09VDS2W1E8]].\nThe *annihilator* of $M$ is the set\n$$\\operatorname{Ann}_R(M) = \\set{r \\in R : rm = 0 \\text{ for all } m \\in M}.$$\nFor an element $m \\in M$, the annihilator of $m$ is $\\operatorname{Ann}_R(m) = \\set{r \\in R : rm = 0}$.\n\n## Remarks\n\n$\\operatorname{Ann}_R(M)$ and $\\operatorname{Ann}_R(m)$ are [[ideals|0MYSA6MYNA]] of $R$.\nExample: $\\operatorname{Ann}_\\Z(\\Z/6\\Z) = 6\\Z$; multiplication by $r$ kills the class of $1$ exactly when $6 \\mid r$, and then it kills every class.\nNon-example: $3 \\notin \\operatorname{Ann}_\\Z(\\Z/6\\Z)$, since $3 \\cdot 1 = 3 \\ne 0$ in $\\Z/6\\Z$.\n$M$ is a *faithful* module when $\\operatorname{Ann}_R(M) = 0$.\nFor an ideal $J$ of $R$, the annihilator of the module $J$ is the [[ideal quotient|16J9DM8V57]] $(0 : J)$.\n$M$ is naturally a module over $R/\\operatorname{Ann}_R(M)$.\n",
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