upper nilradical

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

It is not hard to see that $\operatorname{Nil}^{*}(R)$ is the largest nil ideal in $R$ . Furthermore, we have that $\operatorname{Nil}_{*}(R)\subseteq\operatorname{Nil}^{*}(R)\subseteq J(R)$ , where $\operatorname{Nil}_{*}(R)$ 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 $\operatorname{Nil}_{*}(R)=\operatorname{Nil}^{*}(R)=\operatorname{Nil}(R)$ , the nilradical of $R$ , consisting of all nilpotent elements.
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If $R$ is left (or right) artinian, then $\operatorname{Nil}_{*}(R)=\operatorname{Nil}^{*}(R)=J(R)$ .

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