prime ideal

Let R be a ring. A two-sided proper idealMathworldPlanetmathPlanetmath 𝔭 of a ring R is called a prime idealMathworldPlanetmathPlanetmath if the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath conditions are met:

  1. 1.

    If I and J are left idealsMathworldPlanetmathPlanetmath and the product of ideals IJ satisfies IJ𝔭, then I𝔭 or J𝔭.

  2. 2.

    If I and J are right ideals with IJ𝔭, then I𝔭 or J𝔭.

  3. 3.

    If I and J are two-sided ideals with IJ𝔭, then I𝔭 or J𝔭.

  4. 4.

    If x and y are elements of R with xRy𝔭, then x𝔭 or y𝔭.

R/𝔭 is a prime ringMathworldPlanetmath if and only if 𝔭 is a prime ideal. When R is commutativePlanetmathPlanetmathPlanetmath with identityPlanetmathPlanetmathPlanetmathPlanetmath, a proper ideal 𝔭 of R is prime if and only if for any a,bR, if ab𝔭 then either a𝔭 or b𝔭. One also has in this case that 𝔭R is prime if and only if the quotient ringMathworldPlanetmath R/𝔭 is an integral domain.

Title prime ideal
Canonical name PrimeIdeal
Date of creation 2013-03-22 11:50:54
Last modified on 2013-03-22 11:50:54
Owner djao (24)
Last modified by djao (24)
Numerical id 15
Author djao (24)
Entry type Definition
Classification msc 16D99
Classification msc 13C99
Related topic MaximalIdeal
Related topic Ideal
Related topic PrimeElement