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 let us call such a domain simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to M. Zafrullah (1978) and J. Brewer-W. Heinzer (2002), we give conditions for a *-ideal to be a unique *-product of pairwise *-comaximal ideals with prime radical and characterize *-URDs. We show that the class of URDs 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 $D+XD_S[X]$ construction.

EL BAGHDADI, S., Gabelli, S., Zafrullah, M. (2008). Unique Representation Domains, II. JOURNAL OF PURE AND APPLIED ALGEBRA, 212, 376-393 [10.1016/j.jpaa.2007.06.003].

Unique Representation Domains, II

GABELLI, Stefania;
2008-01-01

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 let us call such a domain simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to M. Zafrullah (1978) and J. Brewer-W. Heinzer (2002), we give conditions for a *-ideal to be a unique *-product of pairwise *-comaximal ideals with prime radical and characterize *-URDs. We show that the class of URDs 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 $D+XD_S[X]$ construction.
2008
EL BAGHDADI, S., Gabelli, S., Zafrullah, M. (2008). Unique Representation Domains, II. JOURNAL OF PURE AND APPLIED ALGEBRA, 212, 376-393 [10.1016/j.jpaa.2007.06.003].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/118226
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