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  "body": "\n## Summary\n\nThe Krull dimension of a module over a possibly noncommutative ring, in the sense of Gabriel and Rentschler, is an ordinal.\nIt measures how far the lattice of submodules is from satisfying the descending chain condition.\nIt is defined by transfinite recursion: dimension $0$ means nonzero [[Artinian|YQNZTQGP1X]].\nDimension at most $\\alpha$ means that every descending chain of submodules has, from some point on, factors of dimension less than $\\alpha$.\nThe right Krull dimension of a ring is the Krull dimension of the ring as a right module over itself.\nFor commutative [[Noetherian rings|0670T59HGE]] it agrees with the classical [[Krull dimension|MVGBS3J1P7]] defined by chains of prime ideals.\n\n## Explanations\n\n### L1 human\n\nIn a commutative ring, dimension counts the longest chain of prime ideals, but in noncommutative rings prime ideals can be scarce.\nThe Gabriel–Rentschler dimension looks instead at descending chains of submodules.\nArtinian modules, where every descending chain stops, get dimension $0$.\nThe integers get dimension $1$: a descending chain of ideals $\\Z \\supseteq a_1\\Z \\supseteq a_2\\Z \\supseteq \\cdots$ can go on forever, but its factors are finite groups, which have dimension $0$.\nA polynomial ring in two variables gets dimension $2$, and so on, matching the usual dimension.\n",
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    "name": "Krull dimension of a noncommutative ring",
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  "name": "Krull dimension of a noncommutative ring",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/PP1/PP11DV3ZMQ.md",
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