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 formula_1 and sometimes called the "assassin" or "assassinator" of (word play between the notation and the fact that an associated prime is an "annihilator").
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 formula_2 Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes.
Definitions.
A nonzero "R" module "N" is called a prime module if the annihilator formula_3 for any nonzero submodule "N' " of "N". For a prime module "N", formula_4 is a prime ideal in "R".
An associated prime of an "R" module "M" is an ideal of the form formula_4 where "N" is a prime submodule of "M". In commutative algebra the usual definition is different, but equivalent: if "R" is commutative, an associated prime "P" of "M" is a prime ideal of the form formula_6 for a nonzero element "m" of "M" or equivalently formula_7 is isomorphic to a submodule of "M".
In a commutative ring "R", minimal elements in formula_8 (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 "x""n""M" = 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 formula_9 is coprimary with "P". An ideal "I" is a "P"-primary ideal if and only if formula_10; thus, the notion is a generalization of a primary ideal.
Properties.
Most of these properties and assertions are given in starting on page 86.
For the case for commutative Noetherian rings, see also Primary decomposition#Primary decomposition from associated primes.