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  "body": "\n## Statement\n\nLet $R$ be a left [[Artinian ring|20QEAM8AX2]] with identity.\nAssume that $R$ is [[reduced|SZHY975FFX]]: its only [[nilpotent element|0ZB2JG1M9C]] is $0$.\nThen $R$ is a [[semisimple ring|NY25HXA478]].\nMore precisely, $R \\cong D_1 \\times \\cdots \\times D_k$ for some [[division rings|0EA5BTA0CX]] $D_1, \\dots, D_k$.\nIf $R$ is moreover commutative, then $R$ is a finite product of [[fields|04C5ZBG1WT]].\n\n## Remarks\n\nThe [[Jacobson radical|XA0DCNH7WY]] $J$ of a left Artinian ring is [[nilpotent|H7RAXJV9CP]], so each element of $J$ is nilpotent.\nSince $R$ is reduced, $J = 0$.\nA left Artinian ring with zero Jacobson radical is semisimple, and the [[Artin–Wedderburn theorem|R1F271180N]] gives $R \\cong M_{n_1}(D_1) \\times \\cdots \\times M_{n_k}(D_k)$.\nIf some $n_i \\ge 2$, the matrix unit $e_{12}$ in that factor is a nonzero element with square $0$, which is impossible; so every $n_i = 1$.\n\nExample: $\\Z/6\\Z \\cong \\Z/2\\Z \\times \\Z/3\\Z$ is Artinian and reduced, and it is a product of two fields.\nThe Artinian hypothesis is needed: $\\Z$ is reduced but not semisimple, since the ideal $2\\Z$ has no complement in $\\Z$.\nReducedness is needed: $\\Z/4\\Z$ is Artinian, but $2$ is a nonzero nilpotent element, and $\\Z/4\\Z$ is not semisimple.\n\n## Explanations\n\n### L1 human\n\nAn Artinian ring is a ring with a strong finiteness condition on its ideals.\nIf such a ring has no nonzero element whose power is zero, then it breaks up into finitely many pieces, and each piece is a division ring.\nIn the commutative case, each piece is a field.\n\n### L2 human\n\nThe nilpotent elements are the only obstruction to semisimplicity for Artinian rings, because the radical of an Artinian ring is a nilpotent ideal.\nSo for Artinian rings, \"reduced\" implies \"semiprimitive\", which implies \"semisimple\".\nIn commutative algebra this is the statement that a reduced Artinian ring is a finite product of fields.\nIt is used, for example, to describe reduced zero-dimensional Noetherian rings and finite étale algebras over a field.\n",
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  "name": "Artinian reduced rings are semisimple",
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