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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nA subset $S$ of $R$ is *locally nilpotent* if the following holds for every finite subset $F \\subseteq S$.\nThere is an integer $n \\ge 1$ such that every product $f_1 f_2 \\cdots f_n$ of $n$ elements of $F$ is $0$.\nThe *Levitzki radical* $\\operatorname{L}(R)$ of $R$ is the sum of all locally nilpotent [[two-sided ideals|0DDA6CY7GJ]] of $R$.\n\n## Remarks\n\nThe sum of all locally nilpotent ideals is again locally nilpotent, so $\\operatorname{L}(R)$ is the largest locally nilpotent ideal of $R$.\nExample: let $R$ be the ring of upper triangular $2 \\times 2$ matrices over a field $k$ and $N$ its strictly upper triangular matrices.\n$N$ is locally nilpotent since $N^2 = 0$, and $R/N \\cong k \\times k$ has no nonzero nilpotent elements, so $\\operatorname{L}(R) = N$.\nNon-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 $m$.\nEvery 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|WF117EJ2HF]] and the [[upper nilradical|3G2JQEYBEK]].\n",
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    "name": "Levitzki radical",
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  "name": "Levitzki radical",
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