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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].\nA subset $S \\subseteq R$ is *left T-nilpotent* if for every sequence $a_1, a_2, a_3, \\ldots$ of elements of $S$ there is $n \\ge 1$ with $a_1 a_2 \\cdots a_n = 0$.\n$R$ is *left perfect* if the following hold.\n\n1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.\n2. $J$ is left T-nilpotent.\n\n*Right perfect* rings are defined with *right T-nilpotent*, which asks for $a_n \\cdots a_2 a_1 = 0$ instead.\n\n## Remarks\n\nExample: a left Artinian ring is left and right perfect: $R/J$ is semisimple, and $J^n = 0$ for some $n$, so every product of $n$ elements of $J$ is $0$.\nNon-example: $\\Z_{(p)}$ is semilocal, but $J = p\\Z_{(p)}$ is not left T-nilpotent, since the constant sequence $p, p, p, \\ldots$ has all products $p^n \\ne 0$.\nBy [[Bass's theorem P|D2M8BSW720]], $R$ is left perfect if and only if every left $R$-module has a projective cover.\nThis is also equivalent to the descending chain condition on principal right ideals.\nMany authors take one of these equivalent conditions as the definition.\n",
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