Cohen-Kaplansky domains: Integral domains with a finite number of irreducible elements

We define an integral domain R to be a Cohen-Kaplansky domain (CK domain) if every element of R is a finite product of irreducible elements and R has only finitely many nonassociate irreducible elements. The purpose of this paper is to investigate CK domains. Many conditions equivalent to R being a CK domain are given, for example, R is a CK domain if and only if R is a one-dimensional semi-local domain and for each nonprincipal maximal ideal M of R, RM is finite and RM is analytically irreducible, or, if and only if G(R), the group of divisibility of R, is finitely generated and rank G(R)= ¦Max(R)¦. We show that a CK domain is a certain special type of composite or pullback of a subring of a finite homomorphic image of a semilocal PID. Noetherian domains with G(R) finitely generated are also investigated.