The Stacks project

Lemma 10.63.9. Let $R$ be a Noetherian ring. Let $M$ be an $R$-module. The union $\bigcup _{\mathfrak q \in \text{Ass}(M)} \mathfrak q$ is the set of elements of $R$ which are zerodivisors on $M$.

Proof. Any element in any associated prime clearly is a zerodivisor on $M$. Conversely, suppose $x \in R$ is a zerodivisor on $M$. Consider the submodule $N = \{ m \in M \mid xm = 0\} $. Since $N$ is not zero it has an associated prime $\mathfrak q$ by Lemma 10.63.7. Then $x \in \mathfrak q$ and $\mathfrak q$ is an associated prime of $M$ by Lemma 10.63.3. $\square$

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