Unique representation domains, II

Abstract

Given a star operation ∗ of finite type, we call a domain R a ∗ -unique representation domain ( ∗ -URD) if each ∗ -invertible ∗ -ideal of R can be uniquely expressed as a ∗ -product of pairwise ∗ -comaximal ideals with prime radical. When ∗ is the t -operation we call the ∗ -URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19–29] and Brewer–Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999–6010], we give conditions for a ∗ -ideal to be a unique ∗ -product of pairwise ∗ -comaximal ideals with prime radical and characterize ∗ -URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t -spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D + X D S [ X ] construction.