Nagata ring

Summary

A Nagata ring is a Noetherian ring in which integral closures are well behaved. For every prime ideal $\mathfrak{p}$, the integral closure of $R/\mathfrak{p}$ in any finite extension of its fraction field is a finite module. The notion is named after Masayoshi Nagata, who gave examples of Noetherian domains whose integral closures are not finite. Every excellent ring is a Nagata ring, and every finitely generated algebra over a Nagata ring is again a Nagata ring. The finiteness of integral closures is what makes normalization a finite operation in algebraic geometry.