Japanese ring

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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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 $R$ is termed a Japanese ring if the following two conditions hold:

• $R$ is finitely generated over integers or over a field
• If $L$ is a finite extension field of the field of fractions of $R$, then the integral closure of $R$ in $L$ is a finitely generated module over $R$.