Straight domains are locally divided

Abstract

We present a proof that all straight domains are locally divided—thereby answering two open problems posed by Dobbs and Picavet, which appeared in the survey “Open Problems in Commutative Ring Theory” written by Cahen, Fontana, Frisch, and Glaz. In fact, we are able to prove a stronger result: a prime ideal of a domain is straight if and only if it is locally divided.

KEYWORDS:

• CPI-extension
• divided domain
• locally divided domain
• straight domain

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13B21
• 13B99
• 13C13
• 13F05

Acknowledgments

I would like to thank my supervisor David McKinnon for his mentorship, insightful advice, and contagious curiosity. I am thankful toward Jason Bell for carefully reading this paper and for providing numerous insightful comments, suggestions, and examples. I am also thankful toward Thomas Bray for his frequent help and many insightful discussions on mathematical topics, and to Dan Labach for proofreading this paper. I wish to thank the referee for useful suggestions and observations. I am grateful to Tiberiu Dumitrescu for his help in creating the final form of this paper: Shortly after submitting my preprint to Arxiv, Dumitrescu notified me of an error in the first preprint of this paper, which related to an incorrect application of a local-global argument in a lemma. By instead using Proposition 2.1, Dumitrescu was able to remedy this problem and generously sent me his work, along with permission to use it in this document.