Japanese ring: Difference between revisions
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{{integral domain property}} | |||
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* <math>R</math> is finitely generated over integers or over a field | * <math>R</math> is finitely generated over integers or over a field | ||
* If <math>L</math> is a finite extension field of the field of fractions of <math>R</math>, then the integral closure of <math>R</math> in <math>L</math> is a [[finitely generated module]] over <math>R</math>. | * If <math>L</math> is a finite extension field of the field of fractions of <math>R</math>, then the integral closure of <math>R</math> in <math>L</math> is a [[finitely generated module]] over <math>R</math>. | ||
Revision as of 00:29, 17 April 2007
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
Origin
The term Japanese ring was introduced by Grothendieck in 1965 in honour of the contributions of the Japanese school to ring theory.
Definition
Definition with symbols
An integral domain is termed a Japanese ring if the following two conditions hold:
- is finitely generated over integers or over a field
- If is a finite extension field of the field of fractions of , then the integral closure of in is a finitely generated module over .
