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 $A$.

Definition 1

$\mathfrak p$ 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 $x$.

Definition 2

$\mathfrak p$ is an associated prime of $M$ if and only if:

$M$ contains a submodule which is isomorphic to the quotient ring $A/\mathfrak p$.

Notation

The set of all associated primes of $M$ is denoted by:

$\map {\operatorname {Ass}_A } M$

Also see

• Equivalence of Definitions of Associated Prime of Module

Sources

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