Let R be a commutative ring. An element xR is said to be nilpotent if xn=0 for some positive integer n. The set of all nilpotent elements of R is an ideal of R, called the nilradical of R and denoted Nil(R). The nilradical is so named because it is the radicalPlanetmathPlanetmath of the zero idealMathworldPlanetmathPlanetmath.

The nilradical of R equals the prime radical of R, although proving that the two are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath requires the axiom of choiceMathworldPlanetmath.

Title nilradical
Canonical name Nilradical
Date of creation 2013-03-22 12:47:52
Last modified on 2013-03-22 12:47:52
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 13A10
Related topic PrimeRadical
Related topic JacobsonRadical
Defines nilpotent