Let R be a ring. A left idealMathworldPlanetmath (resp., right ideal) I of R is a nonempty subset IR such that:

  • a-bI for all a,bI

  • raI (resp. arI) for all aI and rR

A two-sided ideal is a left ideal I which is also a right ideal. If R is a commutative ring, then these three notions of ideal are equivalentPlanetmathPlanetmath. Usually, the word “ideal” by itself means two-sided ideal.

The name “ideal” comes from the study of number theoryMathworldPlanetmath. When the failure of unique factorizationMathworldPlanetmath in number fields was first noticed, one of the solutions was to work with so-called “ideal numbers” in which unique factorization did hold. These “ideal numbers” were in fact ideals, and in Dedekind domainsMathworldPlanetmath, unique factorization of ideals does indeed hold. The term “ideal number” is no longer used; the term “ideal” has replaced and generalized it.

Title ideal
Canonical name Ideal
Date of creation 2013-03-22 11:49:27
Last modified on 2013-03-22 11:49:27
Owner djao (24)
Last modified by djao (24)
Numerical id 18
Author djao (24)
Entry type Definition
Classification msc 14K99
Classification msc 16D25
Classification msc 11N80
Classification msc 13A15
Classification msc 54C05
Classification msc 54C08
Classification msc 54J05
Classification msc 54D99
Classification msc 54E05
Classification msc 54E15
Classification msc 54E17
Related topic Subring
Related topic PrimeIdeal
Defines left ideal
Defines right ideal
Defines 2-sided ideal
Defines two-sided ideal