Maximal non valuative domains

TL;DR

This paper introduces and characterizes maximal non valuative domains, exploring their properties, limitations on maximal ideals, and conditions relating to pseudo-valuation domains.

Contribution

It defines maximal non valuative domains, provides their characterization, and discusses their properties and connections to pseudo-valuation domains.

Findings

Maximal non valuative domains have at most four maximal ideals.

Conditions are identified under which pseudo-valuation domains are maximal non valuative.

Various properties of these domains are discussed.

Abstract

The notion of maximal non valuative domain is introduced and characterized. An integral domain R is called a maximal non valuative domain if R is not a valuative domain but every proper overring of R is a valuative domain. Maximal non valuative domains have at most four maximal ideals. Various properties of maximal non valuative domains are discussed. Conditions are given under which pseudo-valuation domains and maximal non pseudo-valuation domains are maximal non valuative domains.