Over a Noetherian ring each nonzero module has an associated prime.

Lemma 10.63.7. Let $R$ be a Noetherian ring. Let $M$ be an $R$-module. Then

$M = (0) \Leftrightarrow \text{Ass}(M) = \emptyset .$

Proof. If $M = (0)$, then $\text{Ass}(M) = \emptyset$ by definition. If $M \not= 0$, pick any nonzero finitely generated submodule $M' \subset M$, for example a submodule generated by a single nonzero element. By Lemma 10.40.2 we see that $\text{Supp}(M')$ is nonempty. By Proposition 10.63.6 this implies that $\text{Ass}(M')$ is nonempty. By Lemma 10.63.3 this implies $\text{Ass}(M) \not= \emptyset$. $\square$

Comment #830 by on

Suggested slogan: Modules over Noetherian rings are zero if and only if they have no associated primes.

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