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Concept
Associated prime
Summary
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).
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.
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 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 starting on page 86.
If M' ⊆M, then If in addition M' is an essential submodule of M, their associated primes coincide.
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Ontological neighbourhood
Related concepts (5)
Minimal prime ideal
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note: if I is a prime ideal, then I is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal.
Primary ideal
In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n > 0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals.
Primary decomposition
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by .