Associated prime to a module

Definition

Let $R$ be a commutative unital ring and $M$ be a $R$-module. A prime ideal $P$ of $R$ is said to be associated to $M$ if it satisfies the following equivalent conditions:

• $P$ is the annihilator of an element of $M$
• There is an injective homomorphism $A/P\to M$ of $A$-modules

The set of all primes associated to $M$ is denoted as $Ass_{R}M$.