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. A minimal prime ideal over an ideal I in a Noetherian ring R is precisely a minimal associated prime (also called isolated prime) of ; this follows for instance from the primary decomposition of I. In a commutative artinian ring, every maximal ideal is a minimal prime ideal. In an integral domain, the only minimal prime ideal is the zero ideal. In the ring Z of integers, the minimal prime ideals over a nonzero principal ideal (n) are the principal ideals (p), where p is a prime divisor of n. The only minimal prime ideal over the zero ideal is the zero ideal itself. Similar statements hold for any principal ideal domain. If I is a p-primary ideal (for example, a symbolic power of p), then p is the unique minimal prime ideal over I. The ideals and are the minimal prime ideals in since they are the extension of prime ideals for the morphism , contain the zero ideal (which is not prime since , but, neither nor are contained in the zero ideal) and are not contained in any other prime ideal. In the minimal primes over the ideal are the ideals and . Let and the images of x, y in A. Then and are the minimal prime ideals of A (and there are no others). Let be the set of zero-divisors in A. Then is in D (since it kills nonzero ) while neither in nor ; so . All rings are assumed to be commutative and unital. Every proper ideal I in a ring has at least one minimal prime ideal above it. The proof of this fact uses Zorn's lemma. Any maximal ideal containing I is prime, and such ideals exist, so the set of prime ideals containing I is non-empty.

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés (18)

Cours associés (5)

Séances de cours associées (37)

MATH-482: Algebraic number theory

Algebraic number theory is the study of the properties of solutions of polynomial equations with integral coefficients; Starting with concrete problems, we then introduce more general notions like alg

MATH-317: Galois theory

Ce cours commence par un rappel de la théorie de Galois vue en 2ème année : les extensions de corps et la correspondance de Galois. Ensuite, on présente les applications et approfondissements.

MATH-311: Rings and modules

The students are going to solidify their knowledge of ring and module theory with a major emphasis on commutative algebra and a minor emphasis on homological algebra.

In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero.

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.

Anneaux et modules MATH-311: Rings and modules

Couvre les anneaux, les modules, les champs, les idéaux minimaux et le théorème de Nullstellensatz.

Extensions séparables: Anneaux de Dedekind

Explore les extensions séparables et les anneaux de Dedekind, en se concentrant sur les coefficients et les idéaux premiers.

Décomposition primaire : idéal noéthérien MATH-311: Rings and modules

Couvre la décomposition primaire dans les idéaux noéthériens et les idéaux primaires uniques.