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  "body": "\n## Definition\n\nWork with associative [[rings that need not have an identity|MBK7NFE74S]].\nA class $\\mathcal{R}$ of such rings, closed under isomorphism, is a *radical class* in the sense of Kurosh and Amitsur if the following hold.\n\n1. If $A \\in \\mathcal{R}$ and $B$ is a homomorphic image of $A$, then $B \\in \\mathcal{R}$.\n2. Every ring $A$ has a [[two-sided ideal|0DDA6CY7GJ]] $\\mathcal{R}(A)$ that belongs to $\\mathcal{R}$ and contains every ideal of $A$ that belongs to $\\mathcal{R}$.\n3. For every ring $A$, the quotient $A/\\mathcal{R}(A)$ has no nonzero ideal in $\\mathcal{R}$, that is, $\\mathcal{R}(A/\\mathcal{R}(A)) = 0$.\n\nThe ideal $\\mathcal{R}(A)$ is the *$\\mathcal{R}$-radical* of $A$.\nA ring $A$ is *$\\mathcal{R}$-semisimple* if $\\mathcal{R}(A) = 0$.\n\n## Remarks\n\nExample: the class of nil rings is a radical class.\nHomomorphic images of nil rings are nil.\nThe sum of all nil ideals of $A$ is the largest nil ideal, the [[upper nilradical|3G2JQEYBEK]] $N$.\nIf $I/N$ is a nil ideal of $A/N$, then every $x \\in I$ has a power in $N$, so $I$ is nil and $I/N = 0$.\nNon-example: the class of nilpotent rings satisfies clause 1 but not clause 2, since a ring can have an infinite family of nilpotent ideals whose sum is not nilpotent.\nOther radical classes are the Jacobson radical rings (rings equal to their [[Jacobson radical|XA0DCNH7WY]]), the locally nilpotent rings, which give the [[Levitzki radical|9DD8M9R868]], and the class giving the [[prime radical|WF117EJ2HF]].\n",
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