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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $M$ be an $R$-[[module|09VDS2W1E8]].\nFor $m \\in M$, let $\\operatorname{Ann}_R(m) = \\set{r \\in R : rm = 0}$ be the [[annihilator|HB4XDKJTW8]] of $m$.\nA [[prime ideal|06HC6HK0Z8]] $P$ of $R$ is an *associated prime* of $M$ if $P = \\operatorname{Ann}_R(m)$ for some $m \\in M$.\nThe set of associated primes of $M$ is written $\\operatorname{Ass}_R(M)$.\n\n## Remarks\n\nFor an ideal $I$ of $R$, the associated primes of $I$ usually means $\\operatorname{Ass}_R(R/I)$.\nExample: for $M = \\Z/12\\Z$ over $\\Z$, $\\operatorname{Ann}(6) = (2)$ and $\\operatorname{Ann}(4) = (3)$, so $(2)$ and $(3)$ are associated primes; in fact $\\operatorname{Ass}(M) = \\set{(2), (3)}$.\nNon-example: $(0)$ is not an associated prime of $\\Z/12\\Z$, because $12$ kills every element, so no annihilator is $(0)$.\nFor $M = \\Z$, the annihilator of every nonzero element is $(0)$, so $\\operatorname{Ass}(\\Z) = \\set{(0)}$.\nEquivalently, $P \\in \\operatorname{Ass}_R(M)$ when $M$ has a submodule isomorphic to $R/P$.\nFor rings that are not Noetherian, some authors use other variants, such as weakly associated primes.\n\n## Explanations\n\n### L1 human\n\nAn associated prime of a module is a prime ideal that is exactly the set of ring elements killing some one element of the module.\n",
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