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  "body": "\n## Definition\n\nLet $A$ be an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$.\nThe domain $A$ is *N-2* if, for every [[finite extension|0C1Y04SZ1W]] $L$ of $K$, the [[integral closure|G0V86SJBQW]] of $A$ in $L$ is a [[finitely generated|05QY1VEBYD]] $A$-module.\nAn N-2 domain is also called *Japanese*.\nA commutative ring $R$ is a *Nagata ring* if:\n\n1. $R$ is a [[Noetherian ring|0670T59HGE]].\n2. For every [[prime ideal|00MHHZ9SNE]] $\\mathfrak{p}$ of $R$, the domain $R/\\mathfrak{p}$ is N-2.\n\n## Remarks\n\nExample: $\\Z$ is a Nagata ring.\nIt is Noetherian, and its quotients by primes are $\\Z$ and the fields $\\F_p$.\nA field $F$ is N-2, because the integral closure of $F$ in a finite extension $L$ is $L$, an $F$-module of finite dimension.\nThe integral closure of $\\Z$ in a number field $L$ is the ring of integers of $L$, a free $\\Z$-module of rank $[L : \\Q]$.\n\nNon-example: there are one-dimensional Noetherian local domains whose integral closure in their own field of fractions is not a finitely generated module.\nSuch a domain fails clause 2 at the prime $\\mathfrak{p} = 0$.\n\nA theorem of Nagata says that a Noetherian ring $R$ is a Nagata ring if and only if every finitely generated $R$-algebra that is a domain is N-2.\nSuch rings are also called universally Japanese.\nOlder texts call Nagata rings pseudo-geometric rings.\n",
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