integral domain

An integral domain, or domain, is a commutativePlanetmathPlanetmathPlanetmath cancellation ring with an identity elementMathworldPlanetmath 10.

Integral domains are sometimes allowed to be noncommutative, but we adopt the convention that an integral domain is commutative unless otherwise specified.

This notion has essentially nothing to do with the domain of a function ( It is also not very closely related to the notion of integralPlanetmathPlanetmath, which is applied to ring elements, or that of integral closure, which is applied to extensionsPlanetmathPlanetmath of rings, although these concepts are normally applied to integral domains. An integral domain shares some of the properties of the integers (more than other kinds of rings, but by no means all those of interest). Integral domains have fraction fields, which play the role of the rational numbersPlanetmathPlanetmathPlanetmath, and they each have a characteristicPlanetmathPlanetmathPlanetmath (which is either a prime numberMathworldPlanetmath or zero).

Title integral domain
Canonical name IntegralDomain
Date of creation 2013-03-22 11:50:24
Last modified on 2013-03-22 11:50:24
Owner djao (24)
Last modified by djao (24)
Numerical id 16
Author djao (24)
Entry type Definition
Classification msc 13G05
Synonym domain
Related topic CancellationRing
Related topic ZeroDivisor
Related topic WhyEuclideanDomains