upper nilradical

The upper nilradical Nil*(R) of R is the sum (http://planetmath.org/SumOfIdeals) of all (two-sided) nil ideals in R. In other words, aNil*R iff a can be expressed as a (finite) sum of nilpotent elements.

It is not hard to see that Nil*(R) is the largest nil ideal in R. Furthermore, we have that Nil*(R)Nil*(R)J(R), where Nil*(R) is the lower radicalPlanetmathPlanetmath or prime radical of R, and J(R) is the Jacobson radicalMathworldPlanetmath of R.


  • If R is commutativePlanetmathPlanetmathPlanetmath, then Nil*(R)=Nil*(R)=Nil(R), the nilradical of R, consisting of all nilpotent elements.

  • If R is left (or right) artinian, then Nil*(R)=Nil*(R)=J(R).

Title upper nilradical
Canonical name UpperNilradical
Date of creation 2013-03-22 17:29:06
Last modified on 2013-03-22 17:29:06
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Definition
Classification msc 16N40