  Starting with a commutative ring $R$ and an ideal $I$, it is possible to
define a family of rings $R(I)_{a,b}$, with $a,b \in R$, as quotients of the
Rees algebra $\oplus_{n \geq 0} I^nt^n$; among the rings appearing in this
family we find Nagata's idealization and amalgamated duplication. Many
properties of these rings depend only on $R$ and $I$ and not on $a,b$; in this
paper we show that the Gorenstein and the almost Gorenstein properties are
independent of $a,b$. More precisely, we characterize when the rings in the
family are Gorenstein, complete intersection, or almost Gorenstein and we find
a formula for the type.
