We study some properties of a family of rings that are obtained as quotients of the Rees algebra associated with a ring R and an ideal I. We give a complete description of the spectrum of every member of the family and describe the localizations at a prime ideal. Consequently, we are able to characterize the Cohen-Macaulay and Gorenstein properties, generalizing known results stated in the local case. Moreover, we study when these rings are integral domains, reduced, quasi-Gorenstein, or satisfy Serre's conditions.
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