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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $M$ be a [[finitely generated|0XHM30WBVB]] $R$-module.\nThen the set $\\operatorname{Ass}_R(M)$ of [[associated primes|1RJJ33KKSS]] of $M$ is finite.\n\n## Remarks\n\nThe proof builds a chain of submodules $0 = M_0 \\subsetneq M_1 \\subsetneq \\cdots \\subsetneq M_n = M$ with $M_i / M_{i-1} \\cong R/P_i$ for prime ideals $P_i$.\nEach step picks an associated prime of $M/M_{i-1}$, which exists when this module is nonzero, and the chain stops because $M$ is a [[Noetherian module|AK1ZRCXR32]].\nThen $\\operatorname{Ass}_R(M) \\subseteq \\set{P_1, \\dots, P_n}$, because associated primes of an extension lie among those of the two pieces.\nFor example, $\\operatorname{Ass}_\\Z(\\Z/n\\Z)$ is the finite set of ideals $(p)$ for the primes $p$ dividing $n \\ge 2$.\nFinite generation is needed: the $\\Z$-module $\\bigoplus_p \\Z/p\\Z$, over all primes $p$, has every $(p)$ as an associated prime.\n\n## Explanations\n\n### L1 human\n\nA finitely generated module over a Noetherian ring has only finitely many associated primes.\nSo it is \"supported\" by finitely many basic primes, like an integer with finitely many prime factors.\n\n### L2 human\n\nAssociated primes of a nonzero module also exist, so $\\operatorname{Ass}_R(M)$ is finite and nonempty when $M \\ne 0$.\nWith the result that [[zero divisors are the union of associated primes|S37DQGW3TZ]], this lets one use [[prime avoidance|FMP5AWG6PR]] to find non-zero-divisors outside finitely many primes.\nThis is a key step in building regular sequences and in the theory of depth.\n",
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