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  "body": "\n## Summary\n\nA Nagata ring is a [[Noetherian ring|0670T59HGE]] in which [[integral closures|G0V86SJBQW]] are well behaved.\nFor every prime ideal $\\mathfrak{p}$, the integral closure of $R/\\mathfrak{p}$ in any finite extension of its fraction field is a finite module.\nThe notion is named after Masayoshi Nagata, who gave examples of Noetherian domains whose integral closures are not finite.\nEvery [[excellent ring|YF7YFQ58QJ]] is a Nagata ring, and every finitely generated algebra over a Nagata ring is again a Nagata ring.\nThe finiteness of integral closures is what makes normalization a finite operation in algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nStarting from a domain, one often wants to add all elements of the fraction field that are roots of monic polynomials, to make the ring \"normal\".\nFor the integers inside a number field this adds only finitely many new generators, and the result is a finite module over $\\Z$.\nFor some Noetherian domains, though, infinitely many new generators are needed.\nNagata rings are the rings where this never happens, not even after passing to quotients by primes or to finite field extensions.\n",
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