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
and sometimes called the assassin or assassinator of M (word play between the notation and the fact that an associated prime is an annihilator).
[1] In commutative algebra, associated primes are linked to the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings.
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,
where N is a prime submodule of M. In commutative algebra the usual definition is different, but equivalent:[4] if R is commutative, an associated prime P of M is a prime ideal of the form
is isomorphic to a submodule of M. In a commutative ring R, minimal elements in
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
; thus, the notion is a generalization of a primary ideal.
Most of these properties and assertions are given in (Lam 1999) starting on page 86.
For the case for commutative Noetherian rings, see also Primary decomposition#Primary decomposition from associated primes.