In this note, we introduce the notion of Gorenstein algebras. Let R be a commutative Gorenstein ring and A a noetherian R-algebra. We call A a Gorenstein R-algebra if A has Gorenstein dimension zero as an R-module (see [2]), add(D(AA)) = PA, where D = HomR(−, R), and Ap is projective as an Rpmodule for all p ∈ Spec R with dim Rp < dim R. Note that if dim R = ∞ then a Gorenstein R-algebra A is p...

Let (R,m) be a commutative Noetherian local ring. We establish some bounds for the sequence of Bass numbers and their dual for a finitely generated R-module. Introduction Throughout this paper, (R,m, k) is a non-trivial commutative Noetherian local ring with unique maximal ideal m and residue field k. Several authors have obtained results on the growth of the sequence of Betti numbers {βn(k)} (...

We introduce the class of “right almost V-rings” which is properly between the classes of right V-rings and right good rings. A ring R is called a right almost V-ring if every simple R-module is almost injective. It is proved that R is a right almost V-ring if and only if for every R-module M, any complement of every simple submodule of M is a direct summand. Moreover, R is a right almost V-rin...

Let be a commutative Noetherian ring and let I be a proper ideal of . D’Anna and Fontana in [6] introduced a new construction of ring,  named amalgamated duplication of along I. In this paper by considering the ring homomorphism , it is shown that if , then , also it is proved that if , then there exists  such that . Using this result it is shown that if is generically Cohen-Macaulay (resp. gen...