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  "body": "\n## Summary\n\nA semiprimary ring is a ring $R$ such that $R/J(R)$ is semisimple and the [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is nilpotent.\nEvery left Artinian ring is semiprimary, because its [[Jacobson radical is nilpotent|H7RAXJV9CP]].\nSemiprimary rings are left and right [[perfect|49Y9Q949PK]].\nThey are the setting of the [[Hopkins–Levitzki theorem|W29AA0266Z]]: a semiprimary ring is left Artinian exactly when it is left Noetherian.\n\n## Explanations\n\n### L1 human\n\nA semiprimary ring is semisimple up to a radical that dies after finitely many multiplications.\nThis is the structure of a left Artinian ring, but without any chain condition.\nFor example, let $R$ be the ring $k[x_1, x_2, \\ldots]$ with all products $x_i x_j$ set equal to $0$.\nIt is semiprimary: its radical is spanned by the $x_i$, it squares to zero, and the quotient is the field $k$.\nThis ring is not Artinian, because the ideals $(x_n, x_{n+1}, \\ldots)$ form an infinite descending chain.\n",
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