Definition:Associated Prime of Module/Definition 1

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 $M$ if and only if:

$\exists x \in M : \map {\operatorname {Ann}_A} x = \mathfrak p$

where $\map {\operatorname {Ann}_A} x$ is the annihilator of $x$.

That is:

$\exists x \in M : \set {a \in A : a x = 0} = \mathfrak p$

Sources

• 1980: Hideyuki Matsumura: Commutative Algebra Chapter 3: Associated Primes