Maximal Annihilator of Module is Associated Prime

Theorem

Let $A$ be a commutative ring with unity.

Let $M$ be a module over $A$.

Let $\mathbf p$ be a maximal element of the set:

$\set { \map {\operatorname {Ann}_A} x : x \in M , x \ne 0 }$

with respect to the subset relation.

Then $\mathfrak p$ is an associated prime of $M$.

Proof

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