A New Generalization of Cohen-Kaplansky Domains

Abstract

The goal of this thesis is to provide an new generalization of Cohen-Kaplansky domains, stemming from questions related to valuation domains. Recall that a Cohen-Kaplansky domain is an atomic integral domain that contains only a nite number of irreducible elements (up to units). In the new generalization presented in this thesis, we remove the atomic condition required in the de nition of a Cohen-Kaplansky domain and add in the extra condition that our integral domain has nitely many irreducible elements, say 1; 2; ; n, such that for every nonzero nonunit y in the domain there exists an irreducible element, say i with 1 i n, such that i j y.