**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# Integral domain

Summary

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.
"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.
Some sources, notably Lang, use the term entire ring for integral domain.
Some specific kinds of integral domains are given with the following chain of class inclusions:
An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Equivalently:
An integral domain is a nonzero commutative ring with no nonzero zero divisors.
An integral domain is a commutative ring in which the zero ideal {0} is a prime ideal.
An integral domain is a nonzero commutative ring for which every nonzero element is cancellable under multiplication.
An integral domain is a ring for which the set of nonzero elements is a commutative monoid under multiplication (because a monoid must be closed under multiplication).
An integral domain is a nonzero commutative ring in which for every nonzero element r, the function that maps each element x of the ring to the product xr is injective. Elements r with this property are called regular, so it is equivalent to require that every nonzero element of the ring be regular.
An integral domain is a ring that is isomorphic to a subring of a field.

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 concepts (132)

Related courses (9)

Related lectures (99)

Integral domain

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c. "Integral domain" is defined almost universally as above, but there is some variation.

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.

Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. A ring is a set equipped with two binary operations, i.e. operations combining any two elements of the ring to a third.

MATH-310: Algebra

This is an introduction to modern algebra: groups, rings and fields.

MATH-201: Analysis III

Calcul différentiel et intégral.
Eléments d'analyse complexe.

MATH-203(b): Analysis III

Le cours étudie les concepts fondamentaux de l'analyse vectorielle et l'analyse de Fourier en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique.

Rings and ModulesMATH-311: Algebra IV - rings and modules

Covers rings, modules, fields, minimal ideals, and the Nullstellensatz theorem.

Field of Fractions: Definition and PropertiesMATH-110(a): Advanced linear algebra I

Covers the definition and properties of a field of fractions.

Properties of Euclidean DomainsMATH-310: Algebra

Covers the properties of Euclidean domains and irreducible elements in polynomial rings.