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  "body": "\n## Summary\n\nA left perfect ring is a ring $R$ such that $R/J(R)$ is semisimple and the [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is left T-nilpotent.\nBy [[Bass's theorem P|D2M8BSW720]], these are exactly the rings over which every left module has a [[projective cover|WGYXR97NPK]].\nLeft perfect rings are [[semiperfect|ZRC2S3ECFK]], and [[semiprimary rings|XVP517QRH5]], in particular left Artinian rings, are left and right perfect.\nLeft perfect and right perfect are different conditions.\n\n## Explanations\n\n### L1 human\n\nIn a semiperfect ring the finitely generated modules have projective covers, a \"best\" projective module mapping onto them.\nA perfect ring asks this of all modules, however large.\nThis needs a strong smallness condition on the Jacobson radical, called T-nilpotence.\nIt says that for every infinite sequence of elements of the radical, some initial product is already zero.\n",
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