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 $R$.
• If $R$ is commutative, then $\operatorname{Nil}_{*}(R)=\operatorname{Nil}^{*}(R)=\operatorname{Nil}(R)$, the nilradical of $R$, consisting of all nilpotent elements.
• If $R$ is left (or right) artinian, then $\operatorname{Nil}_{*}(R)=\operatorname{Nil}^{*}(R)=J(R)$.