Criteria for unique factorization in integral domains

Let R be an integral domain. In this paper, we introduce a sequence of factorization properties which are weaker than the classical UFD criteria. We give several examples of atomic nonfactorial monoids which satisfy these conditions, but show for several classes of integral domains of arithmetical interest that these factorization properties force unique factorization. In particular, we show that if R satisfies any of our properties and is a Krull domain with finite divisor class group, a nonmaximal order in an algebraic number field, or a generalized Cohen-Kaplansky domain, then R in fact must be factorial. (C) 1998 Elsevier Science B.V. All rights reserved.