Let R be an integral domain. An atom is a nonzero nonunit x of R where x = yz implies that either y or z is a unit. We say that R is an atomic domain if each nonzero nonunit is a finite product of atoms. An atomic domain with only finitely many nonassociate atoms is called a Cohen-Kaplansky (CK) domain. We will investigate atoms in integral domains R with a unique maximal ideal M. Of particular interest will be atoms that are not in M^2.
After studying the atoms in integral domains, we will narrow our focus to CK domains with a unique maximal ideal M. In this pursuit, we investigate atoms in M^2 for these CK domains. We will show that the minimal number of atoms needed to have an atom in M^2 is exactly eight. This disproves a conjecture given by Cohen and Kaplansky in 1946 that the minimal number would be ten. We then classify complete local CK domains with exactly three atoms.