---
tag: ERFZ6S8X0X
kind: concept
name: Associated prime
msc: ["13C05", "13A15"]
specializes: [00MHHZ9SNE]
sources:
  - url: https://en.wikipedia.org/wiki/Associated_prime
same_as:
  wikidata: Q4809264
  wikipedia: Associated prime
---

## Summary

An associated prime of a module $M$ over a commutative ring is a prime ideal that is the [[annihilator|HB4XDKJTW8]] of a single element of $M$.
For a finitely generated module over a [[Noetherian ring|0670T59HGE]], there are only finitely many, and their union is the set of zero divisors on $M$.
When $R$ is Noetherian, the associated primes of $R/I$ are the primes that appear in a [[primary decomposition|ZD34THCXQ0]] of the ideal $I$.

## Explanations

### L1 human

Take one element $m$ of a module and collect all ring elements that kill it.
Sometimes this set is a prime ideal; such primes are the associated primes.
For $\Z/12\Z$, the element $6$ is killed exactly by the even numbers, and the element $4$ exactly by the multiples of $3$.
So $(2)$ and $(3)$ are the associated primes of $\Z/12\Z$, the primes dividing $12$.