Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we invest...

We introduce the notion of strongly ω -Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective (injective) modules and ω-Gorenstein modules. We investigate some characterizations of strongly ω -Gorenstein modules. Consequently, some properties under change of rings are obtained. Keywords—faithfully balanced se...

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f -vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f -vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f -vectors of flag homology spheres and the f -vectors of Gorenstein* order comp...

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely-generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely-generated Gorenstein-projective modules.

We characterize the cd-indices of Gorenstein* posets of rank 5, equivalently the flag f -vectors of Gorenstein* order complexes of dimension 3. As a corollary, we characterize the f -vectors of Gorenstein* order complexes in dimensions 3 and 4. This characterization rise a speculated intimate connection between the f -vectors of flag homology spheres and the f -vectors of Gorenstein* order comp...

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...

Beck et. al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result, that is, we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et. al. also gave a method to construct an infinite family of...

we investigate the relative cohomology and relative homology theories of $f$-gorenstein modules, consider the relations between classical and $f$-gorenstein (co)homology theories.

‎Let $C$ be a semidualizing module‎. ‎We first investigate the properties of‎ ‎finitely generated $G_C$-projective modules‎. ‎Then‎, ‎relative to $C$‎, ‎we introduce and study the rings over which‎ ‎every submodule of a projective (flat) module is $G_C$-projective (flat)‎, ‎which we call $C$-Gorenstein (semi)hereditary rings‎. ‎It is proved that every $C$-Gorenstein hereditary ring is both cohe...

It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov-Lazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noether...