Atomic Pseudo-Valuation Domains

TL;DR

This paper establishes the equivalence between atomic pseudo-valuation domains and boundary valuation domains, providing new characterizations and conditions for identifying pseudo-valuation domains in algebraic structures.

Contribution

It proves the equivalence of atomic pseudo-valuation domains and boundary valuation domains, and characterizes their congruence lattices using power series rings and divisibility conditions.

Findings

Atomic pseudo-valuation domains coincide with boundary valuation domains.

Power series rings provide examples with characterizable congruence lattices.

A sufficient condition on the divisibility group guarantees a domain is pseudo-valuation.

Abstract

Pseudo-valuation domains have been studied since their introduction in 1978 by Hedstrom and Houston. Related objects, boundary valuation domains, were introduced by Maney in 2004. Here, it is shown that the class of atomic pseudo-valuation domains coincides with the class of boundary valuation domains. It is also shown that power series rings and generalized power series rings give examples of pseudo-valuation domains whose congruence lattices can be characterized. The paper also introduces, and makes use of, a sufficient condition on the group of divisibility of a domain to guarantee that it is a pseudo-valuation domain.