upper nilradical
The upper nilradical ${\mathrm{Nil}}^{*}(R)$ of $R$ is the sum (http://planetmath.org/SumOfIdeals) of all (twosided) nil ideals in $R$. In other words, $a\in Ni{l}^{*}R$ iff $a$ can be expressed as a (finite) sum of nilpotent elements.
It is not hard to see that ${\mathrm{Nil}}^{*}(R)$ is the largest nil ideal in $R$. Furthermore, we have that ${\mathrm{Nil}}_{*}(R)\subseteq {\mathrm{Nil}}^{*}(R)\subseteq J(R)$, where ${\mathrm{Nil}}_{*}(R)$ is the lower radical^{} or prime radical of $R$, and $J(R)$ is the Jacobson radical^{} of $R$.
Remarks.

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If $R$ is commutative^{}, then ${\mathrm{Nil}}_{*}(R)={\mathrm{Nil}}^{*}(R)=\mathrm{Nil}(R)$, the nilradical of $R$, consisting of all nilpotent elements.

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If $R$ is left (or right) artinian, then ${\mathrm{Nil}}_{*}(R)={\mathrm{Nil}}^{*}(R)=J(R)$.
Title  upper nilradical 

Canonical name  UpperNilradical 
Date of creation  20130322 17:29:06 
Last modified on  20130322 17:29:06 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  4 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16N40 