Atoms in Quasilocal Integral 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 $M^2$.