##  [Associated Prime](/associated-prime) 

  ##  [Associated Prime](https://algebra.quantumdictionary.io/associated-prime-0) 

  

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**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An associated prime of an R-module M is a prime ideal p that equals the annihilator of some element m ∈ M, i.e., p = Ann_R(m). Equivalently, p is the annihilator of a cyclic submodule R·m and appears as a prime occurring in primary decompositions and the set Ass_R(M).