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. 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 [2] 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. [3]
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: [4] 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 1999) starting on page 86.
For the case for commutative Noetherian rings, see also Primary decomposition#Primary decomposition from associated primes.
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. 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 [2] 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. [3]
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: [4] 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 1999) starting on page 86.
For the case for commutative Noetherian rings, see also Primary decomposition#Primary decomposition from associated primes.