A new characterization of principal ideal domains

TL;DR

This paper offers a refined characterization of principal ideal domains by weakening previous conditions, improves existing theorems, and explores relationships among various domain classes in ring theory.

Contribution

It provides a new characterization of principal ideal domains with weaker assumptions and extends understanding of related domain classes.

Findings

Weaker conditions suffice to characterize principal ideal domains.

Every PC domain is AP, linking these classes.

PC domains are incomparable with pre-Schreier domains.

Abstract

In 2008 N.~Q.~Chinh and P.~H.~Nam characterized principal ideal domains as integral domains that satisfy the follo\-wing two conditions: (i) they are unique factorization domains, and (ii) all maximal ideals in them are principal. We improve their result by giving a characterization in which each of these two conditions is weakened. At the same time we improve a theorem by P.~M.~Cohn which characterizes principal ideal domains as atomic B\'ezout domains. We will also show that every PC domain is AP and that the notion of PC domains is incomparable with the notion of pre-Schreier domains (hence with the notions of Schreier and GCD domains as well).