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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $M$ be an $R$-[[module|09VDS2W1E8]].\nAn element $r \\in R$ is a zero divisor on $M$ if $rm = 0$ for some nonzero $m \\in M$.\nThen the set of zero divisors on $M$ is the union of the [[associated primes|1RJJ33KKSS]] of $M$.\n\n## Remarks\n\nIf $P = \\operatorname{Ann}(m)$ is associated, then each $r \\in P$ kills $m \\ne 0$, so $r$ is a zero divisor.\nConversely, if $rm = 0$ with $m \\ne 0$, the set of [[annihilators|HB4XDKJTW8]] of nonzero elements that contain $\\operatorname{Ann}(m)$ has a maximal member, since $R$ is Noetherian.\nA maximal annihilator $\\operatorname{Ann}(m')$ is prime: if $ab \\in \\operatorname{Ann}(m')$ and $b m' \\ne 0$, then $\\operatorname{Ann}(b m') \\supseteq \\operatorname{Ann}(m') + (a)$, so $a \\in \\operatorname{Ann}(m')$.\nFor example, an integer $r$ is a zero divisor on $\\Z/12\\Z$ if and only if $\\gcd(r, 12) \u003e 1$, that is, $r \\in (2) \\cup (3)$.\nTaking $M = R$, the set of [[zero divisors|0K26YKNBSA]] of $R$, together with $0$, is a finite union of prime ideals.\n\n## Explanations\n\n### L1 human\n\nThe elements of the ring that kill some nonzero element of the module are exactly the elements of the associated primes.\nSo the zero divisors, which form a messy set in general, are a union of a few prime ideals.\n\n### L2 human\n\nCombine it with [[finiteness of associated primes|XR0WZJAM0R]] and [[prime avoidance|FMP5AWG6PR]].\nThis shows: if an ideal $I$ consists of zero divisors on a finitely generated module $M$, then $I$ kills a nonzero element of $M$.\nThis is the basic step in the theory of regular sequences and depth.\nIt is also used to show that $R$ has a non-zero-divisor in an ideal not contained in any associated prime of $R$.\n",
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