A CONSTRUCTION OF QUASI-GORENSTEIN RINGS

Abstract

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 (S n ) and I 𝔭 is a maximal Cohen–Macaulay R 𝔭 -module for every 𝔭 ∈ Spec (R), then R ⋈ I satisfies Serre's condition (S n ). Moreover if R ⋈ I satisfies Serre's condition (S n ), 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. Algebra306 (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 [Formula: see text] 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.