Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper, we study the amalgamated duplication ring R I which is introduced by D'Anna and Fontana. It is shown that if R satisfies Serre's condition (Sn) and Ip is a maximal CohenMacaulay Rp -module for every p ∈ Spec (R), then R I satisfies Serre's condition (Sn). Moreover if R I satisfies Serre's condition (Sn), then so does R. This gives a generalization of the same result for Cohen-Macaulay rings in [D'Anna, A construction of Gorenstein rings, J. Algebra 306 (2006) 507-519]. In addition it is shown that if R is a local ring and Ann R (I) = 0, then R I is quasi-Gorenstein if and only if b R satisfies Serre's condition (S 2 ) and I is a canonical ideal of R. This result improves the result of D'Anna which is corrected by Shapiro and states that if R is a Cohen-Macaulay local ring, then R I is Gorenstein if and only if the canonical ideal of R exists and is isomorphic to I, provided Ann R (I) = 0.

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor, defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F C -pd, and we use this to give a relation between the F C -pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact, such relations force the semidualizing modules under consideration to be trivial.

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Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper, after recalling briefly the main properties of the amalgamated duplication ring R I which is introduced by D'Anna and Fontana, we restrict our attention to the study of the properties of R I, when I is a semidualizing ideal of R, i.e., I is an ideal of R and I is a semidualizing R-module. In particular, it is shown that if I is a semidualizing ideal and M is a finitely generated R-module, then M is totally I-reflexive as an R-module if and only if M is totally reflexive as an (R I)-module. In addition, it is shown that if I is a semidualizing ideal, then R and I are Gorenstein projective over R I, and every injective R-module is Gorenstein injective as an (R I)-module. Finally, it is proved that if I is a non-zero flat ideal of R, then fd R (M) fd R I (M R (R I)) fd R (M R (R I)), for every R-module M.