Almost discrete valuation domains

2019/12/04 by Anderson, Daniel D., Xing, Shiqi, Zafrullah, Muhammad
#13A15 #13A18 #13F05 #13F25 #13G05 #13J05 #13J10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.02304

Abstract

Let D be an integral domain. Then D is an almost valuation (AV-)domain if for a, b∈ D∖ \0\ there exists a natural number n with aⁿ| bⁿ or bⁿ| aⁿ. AV-domains are closely related to valuation domains, for example, D is an AV-domain if and only if the integral closure D is a valuation domain and D⊆ 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 D a DVR, and quasilocal and local API-domains (i.e., for \a_α\_{α∈ Λ}⊆ D, there exists an n with (\a_αⁿ\_{α∈ Λ}) principal). The structure of complete local AV-domains and API-domains is determined.

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