Abstract
Let D be an integral domain. Then D is an almost valuation (AV)-domain if for there exists a natural number n with or AV-domains are closely related to valuation domains, for example, D is an AV-domain if and only if the integral closure is a valuation domain and 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 a DVR, and quasilocal and local API-domains (i.e., for there exists an n with principal). The structure of complete local AV-domains and API-domains is determined.
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