upper nilradical

The upper nilradical ${\mathrm{Nil}}^{*}(R)$ of $R$ is the sum (http://planetmath.org/SumOfIdeals) of all (two-sided) 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.

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

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