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  "body": "\n## Summary\n\nAn associated prime of a module $M$ over a commutative ring is a prime ideal that is the [[annihilator|HB4XDKJTW8]] of a single element of $M$.\nFor a finitely generated module over a [[Noetherian ring|0670T59HGE]], there are only finitely many, and their union is the set of zero divisors on $M$.\nWhen $R$ is Noetherian, the associated primes of $R/I$ are the primes that appear in a [[primary decomposition|ZD34THCXQ0]] of the ideal $I$.\n\n## Explanations\n\n### L1 human\n\nTake one element $m$ of a module and collect all ring elements that kill it.\nSometimes this set is a prime ideal; such primes are the associated primes.\nFor $\\Z/12\\Z$, the element $6$ is killed exactly by the even numbers, and the element $4$ exactly by the multiples of $3$.\nSo $(2)$ and $(3)$ are the associated primes of $\\Z/12\\Z$, the primes dividing $12$.\n",
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