Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4598127 | Journal of Pure and Applied Algebra | 2008 | 18 Pages |
Abstract
Given a star operation â of finite type, we call a domain R a â-unique representation domain (â-URD) if each â-invertible â-ideal of R can be uniquely expressed as a â-product of pairwise â-comaximal ideals with prime radical. When â is the t-operation we call the â-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 â-ideal to be a unique â-product of pairwise â-comaximal ideals with prime radical and characterize â-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+XDS[X] construction.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Said El Baghdadi, Stefania Gabelli, Muhammad Zafrullah,