Definition:Associated Prime of Module/Definition 2

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:

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

Sources

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