In this note, we characterize the (weak) Gorenstein global dimension for arbitrary associative rings. Also, we extend the well-known Hilbert’s syzygy Theorem to the weak Gorenstein global dimension, and we study the weak Gorenstein homological dimensions of direct product of rings which gives examples of non-coherent rings with finite Gorenstein dimensions > 0 and infinite classical weak dimens...

In this paper, we extend the well-known Hilbert’s syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. There is a remarkable body of evidence supporting this claim. Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So ...

Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, sec...

In this paper, we establish, as a generalization of a result on the classical homological dimensions of commutative rings, an upper bound on the Gorenstein global dimension of commutative rings using the global cotorsion dimension of rings. We use this result to compute the Gorenstein global dimension of some particular cases of trivial extensions of rings and of group rings.

A semi-dualizing module over a commutative noetherian ringA is a finitely generated module C with RHomA(C,C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.

In [Hov02], the second author introduced the Gorenstein projective and Gorenstein injective model structures on R-Mod, the category of R-modules, where R is any Gorenstein ring. These two model structures are Quillen equivalent and in fact there is a third equivalent structure we introduce; the Gorenstein flat model structure. The homotopy category with respect to each of these is called the st...

Let R be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over R. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rin...

Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open pr...

Rings of invariants can have nice homological properties even if they do not have finite global dimension. Watanabe’s Theorem [W] gives conditions when the fixed subring of a commutative ring under the action of a finite group is a Gorenstein ring. The Gorenstein condition was extended to noncommutative rings by a condition explored by Idun Reiten in the 1970s, called k-Gorenstein in [FGR]. Thi...