Almost discrete valuation domains.

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 a n | b n or b n | a n. 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 α n } α ∈ Λ) principal). The structure of complete local AV-domains and API-domains is determined. [ABSTRACT FROM AUTHOR]