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  "body": "\n## Statement\n\nLet $R$ be a left [[Artinian ring|ZV04WWYE2Y]], that is, a ring with identity whose [[left ideals|5N2CJZ47D4]] satisfy the descending chain condition.\nLet $J = J(R)$ be its [[Jacobson radical|XA0DCNH7WY]], the intersection of all maximal left ideals of $R$.\nThen $J$ is a [[nilpotent ideal|C2H26BYWHE]]: there is an integer $n \\ge 1$ with $J^n = 0$.\n\n## Remarks\n\nThe chain $J \\supseteq J^2 \\supseteq J^3 \\supseteq \\cdots$ becomes constant, say $J^n = J^{n+1} = I$, and one shows $I = 0$.\nIf $I \\ne 0$, choose a left ideal $L$ minimal among those with $IL \\ne 0$, and $x \\in L$ with $Ix \\ne 0$.\nMinimality gives $L = Rx$ and, since $I(Ix) = I^2 x = Ix \\ne 0$, also $Ix = Rx$.\nSo $x = yx$ for some $y \\in I \\subseteq J$; as $1 - y$ is a [[unit|019NFGFPR5]], $x = 0$, a contradiction.\nExample: for the ring of upper triangular $2 \\times 2$ matrices over a field, $J$ is the set of strictly upper triangular matrices and $J^2 = 0$.\nThe Artinian hypothesis is needed: the local ring $\\Z_{(p)}$ is Noetherian, but $J = p\\Z_{(p)}$ is not nilpotent, since $p^n \\ne 0$ for all $n$.\n\n## Explanations\n\n### L1 human\n\nIn a ring with the descending chain condition, the Jacobson radical is \"small\" in a strong sense.\nMultiplying enough elements of it together always gives zero.\nSo the radical consists of nilpotent elements, and it is even killed by a fixed power.\n\n### L2 human\n\nThis is a key step in the structure theory of Artinian rings.\nWith the [[Artin–Wedderburn theorem|R1F271180N]], which describes $R/J$ as a product of matrix rings over division rings, it shows that every left Artinian ring is [[semiprimary|XVP517QRH5]].\nIt is also used in the proof of the [[Hopkins–Levitzki theorem|W29AA0266Z]], which says that a left Artinian ring is left Noetherian.\n",
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  "name": "Jacobson radical of an Artinian ring is nilpotent",
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