In abstract algebra, an associated prime of a moduleM over a ringR is a type of prime ideal of R that arises as an annihilator of a submodule of M. The set of associated primes is usually denoted by and sometimes called the assassin or assassinator of . 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 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 for any nonzero submodule N' of N. For a prime module N, is a prime ideal in R. 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: 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 ringR, minimal elements in are called isolated primes while the rest of the 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 integern. A nonzero finitely generated moduleM 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.
Properties
Most of these properties and assertions are given in starting on page 86.
If M' ⊆M, then If in addition M' is an essential submodule of M, their associated primes coincide.
It is possible, even for a commutative local ring, that the set of associated primes of a finitely generated module is empty. However, in any ring satisfying the ascending chain condition on ideals every nonzero module has at least one associated prime.
Any uniform module has either zero or one associated primes, making uniform modules an example of coprimary modules.
For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the spectrum If R is an Artinian ring, then this map becomes a bijection.
Matlis' Theorem: For a commutative Noetherian ring R, the map from the isomorphism classes of indecomposable injective modules to the spectrum is a bijection. Moreover, a complete set of representatives for those classes is given by where denotes the injective hull and ranges over the prime ideals of R.
The following properties all refer to a commutative Noetherian ring R:
Every ideal J is expressible as a finite intersection of primary ideals. The radical of each of these ideals is a prime ideal, and these primes are exactly the elements of In particular, an ideal J is a primary ideal if and only if has exactly one element.
The set theoretic union of the associated primes of M is exactly the collection of zero-divisors on M, that is, elements r for which there exists nonzero m in M with mr =0.
If M is a finitely generated module over R, then there is a finite ascending sequence of submodules
For a module M over R, Furthermore, the set of minimal elements of coincides with the set of minimal elements of In particular, the equality holds if consists of maximal ideals.
A module M over R has finite length if and only if M is finitely generated and consists of maximal ideals.
Let be a ring homomorphism between Noetherian rings and F a B-module that is flat over A. Then, for each A-module E,
Examples
If the associated prime ideals of are the ideals and
