Journal of Mathematical Sciences: Advances and ApplicationsAuthor's: Katie Christensen, Ryan Gipson and Hamid Kulosman
Pages: [17] - [31]
Received Date: June 18, 2019; Revised July 15, 2019
Submitted by:
DOI: http://dx.doi.org/10.18642/jmsaa_7100122075
A well-known theorem by Cohn ([3]) from 1968 characterizes the principal ideal domains (PIDs) as atomic Bézout domains, while a theorem by Chinh and Nam ([1]) from 2008 characterizes them as unique factorization domains all of whose maximal ideals are principal. We give a simple new characterization which implies each of these characterizations. For that purpose we introduce a new type of integral domains (we call them PC domains) and using this notion we characterize the PIDs as atomic PC domains. We discuss the importance of PC domains and find their position in a large implication diagram containing various types of integral domains.
principal ideal domain, unique factorization domain, atomic domain, Bézout domain, GCD domain, Schreier domain, pre-Schreier domain, AP domain, MIP domain, PC domain.
