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∈R, as quotients of the Rees algebra ⊕n≥0Intn; 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.