On almost pseudo-valuation domains
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Let D be an integral domain, and let D be the integral closure of D. We show that if D is an almost pseudo-valuation domain (APVD), then D is a quasi-PrÄufer domain if and only if D/P is a quasi-PrÄufer domain for each prime ideal P of D, if and only if D is a valuation domain. We also show that D(X), the Nagata ring of D, is a locally APVD if and only if D is a locally APVD and D is a PrÄufer domain.
Abstract
Let D be an integral domain, and let D be the integral closure of D. We show that if D is an almost pseudo-valuation domain (APVD), then D is a quasi-PrÄufer domain if and only if D/P is a quasi-PrÄufer domain for each prime ideal P of D, if and only if D is a valuation domain. We also show that D(X), the Nagata ring of D, is a locally APVD if and only if D is a locally APVD and D is a PrÄufer domain.
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