Associated prime

In abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually denoted by .

In commutative algebra, associated primes are linked to the Lasker-Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal J is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with .[1] Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.


A nonzero R module N is called a prime module if the annihilator for any nonzero submodule N' of N. For a prime module N, is a prime ideal in R.[2]

An associated prime of an R module M is an ideal of the form where N is a prime submodule of M. In commutative algebra the usual definition is different, but equivalent:[3] if R is commutative, an associated prime P of M is a prime ideal of the form for a nonzero element m of M or equivalently is isomorphic to a submodule of M.

In a commutative ring R, minimal elements in (with respect to the set-theoretic inclusion) are called isolated primes while the rest of the associated primes (i.e., those properly containing associated primes) are called embedded primes.

A module is called coprimary if xm = 0 for some nonzero m  M implies xnM = 0 for some positive integer n. A nonzero finitely generated module M over a commutative Noetherian ring is coprimary if and only if it has exactly one associated prime. A submodule N of M is called P-primary if is coprimary with P. An ideal I is a P-primary ideal if and only if ; thus, the notion is a generalization of a primary ideal.


Most of these properties and assertions are given in (Lam 2001) starting on page 86.

The following properties all refer to a commutative Noetherian ring R:

such that each quotient Mi/Mi−1 is isomorphic to R/Pi for some prime ideals Pi. Moreover every associated prime of M occurs among the set of primes Pi. (In general not all the ideals Pi are associated primes of M.)



  1. Lam 1999, p. 117, Ex 40B.
  2. Lam 1999, p. 85.
  3. Lam 1999, p. 86.
  4. Matsumura 1970, 7.C Lemma
  5. Cohn, P. M. (2003), Basic Algebra, Springer, Exercise 10.9.7, p. 391, ISBN 9780857294289.
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