Nagata ring: Difference between revisions
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An [[integral domain]] is termed a '''Nagata ring''' if every [[quotient ring]] that is an [[integral domain]] is a [[Japanese ring]]. Nagata rings are also termed '''pseudo-geometric rings''' and '''universally Japanese rings'''. | An [[integral domain]] is termed a '''Nagata ring''' if every [[quotient ring]] that is an [[integral domain]] is a [[Japanese ring]]. Nagata rings are also termed '''pseudo-geometric rings''' and '''universally Japanese rings'''. | ||
Latest revision as of 16:27, 12 May 2008
This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions
History
Origin of the term
Grothendieck dubbed these rings as universally Japanese rings. Nagata, in his paper, used the term pseudo-geometric ring. Nowadays, the term Nagata ring is most commonly used.
Definition
Symbol-free definition
An integral domain is termed a Nagata ring if every quotient ring that is an integral domain is a Japanese ring. Nagata rings are also termed pseudo-geometric rings and universally Japanese rings.