Families of Gorenstein and almost Gorenstein rings

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 ${\oplus }_{n\ge 0}{I}^n{t}^n$; 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.