Associated prime to a module

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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_RM.