nil and nilpotent ideals

An element x of a ring is nilpotentPlanetmathPlanetmath if xn=0 for some positive integer n.

A ring R is nil if every element in R is nilpotent. Similarly, a one- or two-sided idealMathworldPlanetmath is called nil if each of its elements is nilpotent.

A ring R [resp. a one- or two sided ideal A] is nilpotent if Rn=0 [resp. An=0] for some positive integer n.

A ring or an ideal is locally nilpotent if every finitely generatedMathworldPlanetmathPlanetmath subring is nilpotent.

The following implicationsMathworldPlanetmath hold for rings (or ideals):

nilpotentlocally nilpotentnil
Title nil and nilpotent ideals
Canonical name NilAndNilpotentIdeals
Date of creation 2013-03-22 13:13:25
Last modified on 2013-03-22 13:13:25
Owner mclase (549)
Last modified by mclase (549)
Numerical id 6
Author mclase (549)
Entry type Definition
Classification msc 16N40
Related topic KoetheConjecture
Defines nil
Defines nil ring
Defines nil ideal
Defines nil right ideal
Defines nil left ideal
Defines nil subring
Defines nilpotent
Defines nilpotent element
Defines nilpotent ring
Defines nilpotent ideal
Defines nilpotent right ideal
Defines nilpotent left ideal
Defines nilpotent subring
Defines locally nilpotent
Defines locally nilpotent ring
Defines locally nilpo