Prime ideals | Commutative algebra | Module theory

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 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. (Wikipedia).

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Related pages

Prime ideal | Injective module | Annihilator (ring theory) | Primary decomposition | Injective hull | Noetherian module | Radical of an ideal | Commutative algebra | Artinian ring | Ascending chain condition | Free abelian group | Noetherian ring | Primary ideal | Ring (mathematics) | Abstract algebra | Spectrum of a ring | Finitely generated module | Abelian group | Module (mathematics) | Uniform module