Definition:Associated Prime of Module

Definition
Let $A$ be a commutative ring with unity.

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

Let $\mathfrak p$ be a prime ideal in $A$.

$\mathfrak p$ is an associated prime of $M$ :
 * $\exists x \in M : \map {\operatorname {Ann}_A} x = \mathfrak p$

where:
 * $\map {\operatorname {Ann}_A} x := \set {a \in A : a x = 0}$

is the annihilator of $x$.