On domains with properties weaker than atomicity

Abstract

In general, ring theory is focused on atomic rings, i.e., rings in which every non-zero non-unit element has a factorization into irreducible elements. In a recent article of Boynton and Coykendall, the two authors introduce two properties that are slightly weaker than atomicity, which they call “almost atomicity” and “quasi-atomicity”. In this article, we classify various properties weaker than atomicity and find domains that possess certain combinations of them.

Keywords:

• Ring theory; atomicity; Furstenberg domains

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13F15

Acknowledgements

This project began in a University of Georgia VRG group run by Pete L. Clark and Paul Pollack in 2015–2016. The author gratefully acknowledges funding support from the RTG in Algebraic Geometry, Algebra, and Number Theory, and from the National Science Foundation RTG grant DMS-1344994. In addition, the author wishes to thank Pete L. Clark, Jim Coykendall, Daniel Krashen, and Dino Lorenzini for their help in writing this article. The author also wishes to thank the referee for reviewing this article so thoroughly.