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  "body": "\n## Summary\n\nLet $M$ be a module over a commutative ring $R$.\nThe annihilator of $M$ is the set of elements of $R$ that send every element of $M$ to zero.\nIt is an ideal of $R$.\nIt measures how far the action of $R$ on $M$ is from being faithful.\nIt is used to study the [[support of a module|FETMR0M0M8]] and the [[associated primes|ERFZ6S8X0X]].\nIt differs from the [[annihilator of a subspace|0ASHNYF7AG]] in linear algebra, which lives in the dual space.\n\n## Explanations\n\n### L1 human\n\nThe annihilator of a module collects the ring elements that \"kill\" the whole module.\nFor the $\\Z$-module $\\Z/6\\Z$, multiplying by $6$, $12$ or $-18$ gives zero on every element, and multiplying by any integer outside $6\\Z$ does not.\nSo the annihilator of $\\Z/6\\Z$ is $6\\Z$.\nFor a single element $m$, its annihilator is the set of $r$ with $rm = 0$.\n",
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