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- W2993722304 abstract "Let $D$ be an integral domain. Then $D$ is an almost valuation (AV-)domain if for $a, bin Dsetminus {0}$ there exists a natural number $n$ with $a^{n}mid b^{n}$ or $b^{n}mid a^{n}$. AV-domains are closely related to valuation domains, for example, $D$ is an AV-domain if and only if the integral closure $bar{D}$ is a valuation domain and $Dsubseteq bar{D}$ is a root extension. In this note we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with $bar{D}$ a DVR, and quasilocal and local API-domains (i.e., for ${a_{alpha}}_{alphain Lambda}subseteq D$, there exists an $n$ with $({a_{alpha}^{n}}_{alphain Lambda})$ principal). The structure of complete local AV-domains and API-domains is determined." @default.
- W2993722304 created "2019-12-13" @default.
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- W2993722304 date "2019-12-04" @default.
- W2993722304 modified "2023-09-27" @default.
- W2993722304 title "Almost discrete valuation domains" @default.
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