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  "body": "\n## Definition\n\nLet $A$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.\nFor a nonzero [[ideal|0MYSA6MYNA]] $I$ of $A$, let $I^{-1} = \\set{x \\in K : xI \\subseteq A}$ and $I_v = \\set{x \\in K : x I^{-1} \\subseteq A}$.\nThe ideal $I$ is *divisorial* if $I_v = I$.\n$A$ is a *Mori domain* if every strictly ascending chain $I_1 \\subsetneq I_2 \\subsetneq \\cdots$ of nonzero divisorial ideals of $A$ is finite.\n\n## Remarks\n\nEvery nonzero principal ideal $aA$ is divisorial, since $(aA)^{-1} = a^{-1}A$ and then $(aA)_v = aA$.\nExample: every Noetherian domain is a Mori domain, since it has no infinite strictly ascending chain of ideals at all.\nNon-example: the ring $\\overline{\\Z}$ of all algebraic integers is not Mori; the principal ideals $(2^{1/2^n})$ for $n \\ge 1$ form a strictly ascending chain, since $2^{1/2^{n+1}} / 2^{1/2^{n}} = 2^{-1/2^{n+1}}$ is not an algebraic integer.\nEvery [[Krull domain|CB9WJCN9BM]] is a Mori domain, and a Mori domain is Krull if and only if it is completely integrally closed.\n",
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    "name": "Mori domain",
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  "name": "Mori domain",
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