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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nThe *upper nilradical* $\\operatorname{Nil}^*(R)$ of $R$ is the sum of all [[nil ideals|NZ1G9WRWFB]] of $R$.\nThese are the [[two-sided ideals|0DDA6CY7GJ]] $I$ of $R$ such that every element of $I$ is nilpotent.\n\n## Remarks\n\nThe 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.\nHence $\\operatorname{Nil}^*(R)$ is itself a nil ideal, and it is the largest nil ideal of $R$.\nExample: in $\\Z/12\\Z$ the nil ideals are $0$ and $6\\Z/12\\Z$, so $\\operatorname{Nil}^*(\\Z/12\\Z) = \\set{0, 6}$.\nNon-example: in $M_2(\\Q)$ the element $e_{12}$ is nilpotent but does not lie in $\\operatorname{Nil}^*(M_2(\\Q))$.\nIndeed the only ideals of $M_2(\\Q)$ are $0$ and the whole ring, which contains $1$, so $\\operatorname{Nil}^*(M_2(\\Q)) = 0$.\nIn general $\\operatorname{Nil}_*(R) \\subseteq \\operatorname{L}(R) \\subseteq \\operatorname{Nil}^*(R)$, where $\\operatorname{Nil}_*$ is the [[prime radical|WF117EJ2HF]] and $\\operatorname{L}$ the [[Levitzki radical|9DD8M9R868]], and both inclusions can be strict.\nFor a commutative ring the upper nilradical is the [[nilradical|QQDN6EYQNJ]].\n",
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