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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
