Upper nilradical

Remarks

The sum of two nil ideals $I$ and $I'$ is nil, since $(I + I')/I' \cong I/(I \cap I')$ is nil and $I'$ is nil. Hence $\operatorname{Nil}^*(R)$ is itself a nil ideal, and it is the largest nil ideal of $R$. Example: in $\Z/12\Z$ the nil ideals are $0$ and $6\Z/12\Z$, so $\operatorname{Nil}^*(\Z/12\Z) = \set{0, 6}$. Non-example: in $M_2(\Q)$ the element $e_{12}$ is nilpotent but does not lie in $\operatorname{Nil}^*(M_2(\Q))$. Indeed the only ideals of $M_2(\Q)$ are $0$ and the whole ring, which contains $1$, so $\operatorname{Nil}^*(M_2(\Q)) = 0$. In general $\operatorname{Nil}_*(R) \subseteq \operatorname{L}(R) \subseteq \operatorname{Nil}^*(R)$, where $\operatorname{Nil}_*$ is the prime radical and $\operatorname{L}$ the Levitzki radical, and both inclusions can be strict. For a commutative ring the upper nilradical is the nilradical.