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  "body": "\n## Summary\n\nThe upper nilradical of a ring is its largest [[nil ideal|NZ1G9WRWFB]], the sum of all two-sided ideals that consist of nilpotent elements.\nIt contains the [[prime radical|WF117EJ2HF]] and the [[Levitzki radical|9DD8M9R868]].\nFor a [[commutative ring|0310REJJY8]] all three coincide with the [[nilradical|QQDN6EYQNJ]].\nWhether the upper nilradical also contains every nil one-sided ideal is the open [[Köthe conjecture|RT2D9JEVRR]].\n\n## Explanations\n\n### L1 human\n\nAn ideal is nil when each of its elements has some power equal to zero.\nThe upper nilradical gathers all such ideals into one.\nIn a commutative ring this is just the set of nilpotent elements.\nIn a noncommutative ring a nilpotent element need not lie in any nil ideal: in $M_2(\\Q)$ the matrix unit $e_{12}$ is nilpotent, but the only nil ideal is $0$.\n",
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