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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]].\nThe *prime radical* of $R$ is the intersection of all [[prime ideals|E8K2MN5V1E]] of $R$:\n$$\\operatorname{Nil}_*(R) = \\bigcap_{P \\text{ prime}} P.$$\nIt is a [[two-sided ideal|0DDA6CY7GJ]] of $R$.\n\n## Remarks\n\nExample: let $R$ be the ring of upper triangular $2 \\times 2$ matrices over a field $k$ and $N$ its strictly upper triangular matrices.\nThen $R/N \\cong k \\times k$, the prime ideals of $R$ are the kernels of the two maps $R \\to k$ that take a diagonal entry, and their intersection is $N$.\nNon-example: the matrix unit $e_{11}$ is not in the prime radical of $R$, since it is a nonzero idempotent, while every element of the prime radical is nilpotent.\nEquivalent forms: $\\operatorname{Nil}_*(R)$ is the smallest semiprime ideal of $R$, and it is the set of strongly nilpotent elements of $R$.\nIt is also called the lower nilradical or the Baer radical, and it is contained in the [[upper nilradical|3G2JQEYBEK]].\nFor a commutative ring it equals the [[nilradical|QQDN6EYQNJ]].\n",
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