Levitzki radical

Remarks

The sum of all locally nilpotent ideals is again locally nilpotent, so $\operatorname{L}(R)$ is the largest locally nilpotent ideal of $R$. Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $k$ and $N$ its strictly upper triangular matrices. $N$ is locally nilpotent since $N^2 = 0$, and $R/N \cong k \times k$ has no nonzero nilpotent elements, so $\operatorname{L}(R) = N$. Non-example: in $M_2(\Q)$ the set $\set{e_{12}, e_{21}}$ consists of nilpotent elements but is not locally nilpotent, since $(e_{12} e_{21})^m = e_{11} \ne 0$ for all $m$. Every nilpotent ideal is locally nilpotent and every locally nilpotent ideal is nil, which gives $\operatorname{Nil}_*(R) \subseteq \operatorname{L}(R) \subseteq \operatorname{Nil}^*(R)$ with the prime radical and the upper nilradical.