  We study some properties of a family of rings $R(I)_{a,b}$ that are obtained
as quotients of the Rees algebra associated with a ring $R$ and an ideal $I$.
In particular, 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
$R(I)_{a,b}$ is an integral domain, reduced, quasi-Gorenstein, or satisfies
Serre's conditions.
