Set of zero divisors on a module equals union of associated primes

This article defines a result where the base ring (or one or more of the rings involved) is Noetherian
View more results involving Noetherianness or Read a survey article on applying Noetherianness

Statement

In terms of sets

Let $R$ be a Noetherian ring and $M$ be a $R$-module. Then the set:

$ZD(R):=\{r\in R\mid \exists 0\neq m\in M,rm=0\}$

equals the union of all the associated primes to the module $M$.

In terms of elements

Let $R$ be a Noetherian ring and $M$ be a $R$-module. Then, if an element $r\in R$ annihilates some element $m\in M$, we can find an element $m'\in M$ such that $r$ annihilates $m'$, and further, such that the annihilator of $m'$ is a prime ideal (this will turn out to be the associated prime containing $r$).

Related facts

• Noetherian implies every element in minimal prime is zero divisor
• Reduced Noetherian implies zero divisor in minimal prime