**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Prime ideal

Summary

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.
Primitive ideals are prime, and prime ideals are both primary and semiprime.
An ideal P of a commutative ring R is prime if it has the following two properties:
If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P,
P is not the whole ring R.
This generalizes the following property of prime numbers, known as Euclid's lemma: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say
A positive integer n is a prime number if and only if is a prime ideal in
A simple example: In the ring the subset of even numbers is a prime ideal.
Given an integral domain , any prime element generates a principal prime ideal . Eisenstein's criterion for integral domains (hence UFDs) is an effective tool for determining whether or not an element in a polynomial ring is irreducible. For example, take an irreducible polynomial in a polynomial ring over some field .
If R denotes the ring of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve).
In the ring of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even.
In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly two ideals of R, namely M itself and the whole ring R. Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

Related concepts (119)

Related lectures (142)

Related courses (10)

Primary Decomposition: Understanding Schemes

Explores primary decomposition and schemes in algebraic geometry, emphasizing the importance of working over non-algebraically closed fields and the concept of fibers of morphisms.

Dimension Theory of Rings

Explores the dimension theory of rings, focusing on chains of ideals and prime ideals.

Field Properties: Irreducibility and Units

Covers the properties of fields, including irreducibility and units in polynomials.

Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, P is not the whole ring R.

Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal.

C'est un cours introductoire dans la théorie d'anneau et de corps.

Study basic concepts of modern algebra: groups, rings, fields.

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.

Let K be a field with char(K) ≠ 2. The Witt-Grothendieck ring (K) and the Witt ring W (K) of K are both quotients of the group ring ℤ[𝓖(K)], where 𝓖(K) := K*/(K*)2 is the square

We prove that the multiplier algebra of the Drury-Arveson Hardy space H-n(2) on the unit ball in C-n has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to hi

2011