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1985•Canadian Mathematical BulletinOpen access

Integrally Closed Condensed Domains are Bézout

David F. Anderson, Jimmy T. Arnold, David E. Dobbs

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Abstract

Abstract It is proved that an integral domainRis a Bézout domain if (and only if)Ris integrally closed andI J= {ij|i∊I, j ∊J} for all idealsIandJofR; that is, if (and only if) R is an integrally closed condensed domain. The article then introduces a weakening of the "condensed" concept which, in the context of thek+Mconstruction, is equivalent to a certain field-theoretic condition. Finally, the field extensions satisfying this condition are classified.

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Abstract It is proved that an integral domainRis a Bézout domain if (and only if)Ris integrally closed andI J= {ij|i∊I, j ∊J} for all idealsIandJofR; that is, if (and only if) R is an integrally closed condensed domain. The article then introduces a weakening of the "condensed" concept which, in the context of thek+Mconstruction, is equivalent to a certain field-theoretic condition. Finally, the field extensions satisfying this condition are classified.

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Available abstract

Abstract It is proved that an integral domainRis a Bézout domain if (and only if)Ris integrally closed andI J= {ij|i∊I, j ∊J} for all idealsIandJofR; that is, if (and only if) R is an integrally closed condensed domain. The article then introduces a weakening of the "condensed" concept which, in the context of thek+Mconstruction, is equivalent to a certain field-theoretic condition. Finally, the field extensions satisfying this condition are classified.

Key concepts: Integrally closed, Mathematics, Domain (mathematical analysis), Context (archaeology), Field (mathematics), Integral domain, Pure mathematics, Mathematical analysis

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