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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] and let $M$ be a right $R$-[[module|0NZ8JA25WK]].\nThe *Krull dimension* $\\operatorname{Kdim} M$, when it exists, is an [[ordinal|0YSCSX5YZC]] or $-1$, defined by transfinite recursion as follows.\n\n1. $\\operatorname{Kdim} M = -1$ if and only if $M = 0$.\n2. For an ordinal $\\alpha \\ge 0$, $\\operatorname{Kdim} M \\le \\alpha$ if, for every descending chain $M_0 \\supseteq M_1 \\supseteq M_2 \\supseteq \\cdots$ of [[submodules|05WDSN597C]] of $M$, there is $N$ such that $\\operatorname{Kdim}(M_i / M_{i+1}) \u003c \\alpha$ for all $i \\ge N$.\n3. $\\operatorname{Kdim} M = \\alpha$ if $\\operatorname{Kdim} M \\le \\alpha$ and $\\operatorname{Kdim} M \\not\\le \\beta$ for every $\\beta \u003c \\alpha$.\n\nIf no ordinal $\\alpha$ satisfies $\\operatorname{Kdim} M \\le \\alpha$, then $M$ has no Krull dimension.\nThe *right Krull dimension* of $R$ is $\\operatorname{Kdim} R_R$, the Krull dimension of $R$ as a right module over itself.\n\n## Remarks\n\nFrom clause 2 with $\\alpha = 0$, a module has Krull dimension $0$ exactly when it is nonzero and [[Artinian|YQNZTQGP1X]]: the factors of every descending chain are eventually $0$.\n\nExample: $\\operatorname{Kdim} \\Z_\\Z = 1$.\nTake a descending chain $M_0 \\supseteq M_1 \\supseteq \\cdots$ of subgroups of $\\Z$.\nIf every $M_i = a_i\\Z$ is nonzero, every factor $M_i / M_{i+1}$ is finite, hence Artinian.\nOtherwise some $M_i = 0$, and all later factors are $0$.\nSo $\\operatorname{Kdim} \\Z \\le 1$, and $\\Z$ is not Artinian, so $\\operatorname{Kdim} \\Z = 1$.\n\nNon-example: an infinite-dimensional vector space $V$ over a field has no Krull dimension.\nSplitting a basis into infinitely many infinite parts gives a descending chain $V \\supsetneq M_1 \\supsetneq M_2 \\supsetneq \\cdots$ in which every factor is infinite-dimensional.\nSo an induction on $\\alpha$ shows that $\\operatorname{Kdim} V \\le \\alpha$ fails for every ordinal $\\alpha$.\n\nEvery [[Noetherian module|AK1ZRCXR32]] has a Krull dimension.\nFor a commutative Noetherian ring, the Krull dimension defined here equals the classical [[Krull dimension|MVGBS3J1P7]] given by chains of prime ideals.\nThe definition is the deviation of the partially ordered set of submodules of $M$.\n",
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