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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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.
La décomposition primaire est une généralisation de la décomposition d'un nombre entier en facteurs premiers. Cette dernière décomposition, connue depuis Gauss (1832) sous le nom de théorème fondamental de l'arithmétiqueGauss 1832., s'étend naturellement au cas d'un élément d'un anneau principal. Une décomposition plus générale est celle d'un idéal d'un anneau de Dedekind en produit d'idéaux premiers; elle a été obtenue en 1847 par Kummer (dans le formalisme encore peu maniable des « nombres idéaux ») à l'occasion de ses recherches sur le dernier théorème de FermatKummer 1847.
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