Commutative algebra | Algebraic geometry

Nagata ring

In commutative algebra, an N-1 ring is an integral domain whose integral closure in its quotient field is a finitely generated -module. It is called a Japanese ring (or an N-2 ring) if for every finite extension of its quotient field , the integral closure of in is a finitely generated -module (or equivalently a finite -algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N-2 rings). A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring, but this concept is not used much. (Wikipedia).

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levitron

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Field of fractions | Prime ideal | Quotient ring | Integral domain | Formal power series | Subring | Ideal (ring theory) | Algebraic variety | Commutative algebra | Polynomial | Principal ideal domain | Perfect field | Polynomial ring | Field extension | Dedekind domain | Characteristic (algebra) | Field (mathematics) | Algebraic geometry | Noetherian ring | Ring (mathematics) | Discrete valuation ring | Integrally closed domain | Local ring | Completion of a ring | Degree of a field extension | Finitely generated module | Module (mathematics)