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  "body": "\n## Summary\n\nThe prime radical of a ring is the intersection of all its [[prime ideals|E8K2MN5V1E]].\nIt is also called the lower nilradical or the Baer radical.\nFor a [[commutative ring|0310REJJY8]] it is the [[nilradical|QQDN6EYQNJ]].\nIt is the smallest of the classical radicals of a noncommutative ring, and it vanishes exactly for [[semiprime rings|5F0YMCV50R]].\n\n## Explanations\n\n### L1 human\n\nIn a commutative ring the nilpotent elements form an ideal, but in a noncommutative ring they need not.\nThe prime radical is a good replacement for this ideal.\nIt is a two-sided ideal made of nilpotent elements, cut out by the prime ideals.\nFor example, in the ring of upper triangular $2 \\times 2$ matrices over a field, the prime radical is the set of strictly upper triangular matrices.\n",
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