Families of Gorenstein and almost Gorenstein rings (Q517228)

Let \(R\) be a commutative ring, \(I \subset R\) an ideal and \(t\) an indeterminate. For two elements \(a\), \(b \in R\), we put \[ R(I)_{a,b} = A[It]/I^2(t^2+at+b)A[It]. \] Then we obtain a family of commutative rings, which contains interesting rings, for example, the idealization, the amalgamated duplication and so on. In the present paper, authors studied the Gorenstein (resp.\ almost Gorenstein, Cohen-Macaulay, complete intersection) property of \(R(I)_{a,b}\). The gave necessary and sufficient condition for \(R(I)_{a,b}\) to be Gorenstein (almost Gorenstein, e.t.c.). It is surprised that such properties of \(R(I)_{a,b}\) is independent of the choice of \(a\) and \(b\).