Atoms in Quasilocal Integral Domains

TL;DR

This paper studies the structure of atoms in quasilocal integral domains, especially Cohen-Kaplansky domains, revealing new examples and properties of atoms in various parts of the maximal ideal.

Contribution

It provides new insights into the distribution of atoms in quasilocal domains and constructs a counterexample to a Cohen-Kaplansky statement.

Findings

Existence of atoms in M^2 in certain CK domains

Construction of a CK domain with exactly eight atoms including one in M^2

Analysis of atoms outside and inside M^2 in quasilocal domains

Abstract

Let $(R,M)$ be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of $R$. Special attention is given to the set of atoms in $M \backslash M^2$ and to the existence of atoms in $M^2$. While our main interest is in local Cohen-Kaplansky (CK) domains (atomic integral domains with only finitely many non-associate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight nonassociate atoms having an atom in $M^2$.