Let R be a right GF-closed ring with finite left and right Gorenstein global dimension. We prove that ifI is an ideal of R such that R|I is a semi-simple ring, then the Gorenstein at dimension of R|I as a right R-module and the Gorenstein injective dimension of R|I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the Gorenstein at dimension of S equals to the Gorenstein injective dimension of S.

‎The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics‎, ‎where group rings play the main role‎. ‎The group ring $\rga$ is actually an associative ring that inherits the properties of the group $\ga$ and the ring of coefficients $R$‎. ‎In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory‎, ‎it also has applications in algebraic topology‎, ‎homological algebra‎, ‎algebraic K-theory and algebraic coding theory‎.‎In this article‎, ‎we provide a complete description of Gorenstein flat-cotorsion modules over the group ring $\rga$‎,‎where $\ga$ is a group and $R$ is a commutative ring‎. ‎It will be shown that if $\ga'\leqslant \ga$ is a finite-index subgroup‎, ‎then the restriction of scalars along the ring homomorphism $\rga'\rt\rga$ as well as its right adjoint $\rga\otimes_{\rga'}-$‎, ‎preserve the class of Gorenstein flat-cotorsion modules‎. ‎Then‎, ‎as a result‎, ‎Serre's Theorem is proved for the invariant $\Ghcd_{R}\ga$‎, ‎which refines the Gorenstein homological dimension of $\ga$ over $R$‎, ‎$\Ghd_{R}\ga$‎, ‎and is defined using flat-cotorsion modules‎. ‎Moreover‎, ‎we show that the inequality $\GF (\rga)\leqslant \GF (R)+{\cd_{R}\ga}$ holds for the group ring $\rga$‎, ‎where $\GF (R)$ denotes the supremum of Gorenstein flat-cotorsion dimensions of all $R$-modules and $\cd_{R}\ga$ is the cohomological dimension of $\ga$‎ over $R$‎.

Let R be a commutative ring, Γ,be a group, and Γ, ′,,Γ,be a subgroup of finite index. In this paper, we study the class of Gorenstein flat (resp. Gorenstein cotorsion) modules over the group ring RΓ, . In particular, we discuss the relationship between Gorenstein flat (resp. Gorenstein cotorsion) dimension of RΓ, −, modules and that of RΓ, ′, −, modules. We will prove that, for any RΓ, −, module M of finite Gorenstein flat dimension (GfdRΓ, M < 1 ), there is an inequality GfdRΓ, M ⩽,GfdRM + hdRΓ, , where hdRΓ,denotes the homological dimension of Γ,over R. Analogously, we prove that if Γ,is finite, there is an inequality GctdRΓ, M ⩽,GctdRM + cdRΓ, , where GctdRΓ, M (GctdRM) is the Gorenstein cotorsion dimension of module M over RΓ,(resp. over R) and cdRΓ,is the cohomological dimension of Γ,over R.

Let $R$ be an associative ring and let $n$ be a non-negative integer‎. ‎In this paper‎, ‎we consider $n$-super finitely presented‎, ‎Gorenstein $n$-weak injective and Gorenstein $n$-weak flat modules under change of rings‎. For‎ ‎an excellent extension‎ $S\geq R$, ‎w‎e show ‎that the Gorenstein $n$-super finitely presented dimensions (resp‎. ‎weak Gorenstein $n$-super finitely presented dimensions) of rings $R$ and $S$ coincide‎.

Let (R; m) be a commutative noetherian local ring and 􀀀 be a  nite group. It is proved that if R admits a dualiz-ing module, then the group ring R􀀀 has a dualizing bimodule, as well. Moreover, it is shown that a  nitely generated R􀀀-module M has generalized Gorenstein dimension zero if and only if it has generalized Gorenstein dimension zero as an R-module.

Let A be an abelian category with enough projective objects and let X be a full subcategory of A. We define Gorenstein projective objects with respect to X and YX, respectively, where YX={YÎCh (A)|Y is acyclic and ZnYÎX}. We point out that under certain hypotheses, these two Gorensein projective objects are related in a nice way. In particular, if P (A) ÍX, we show that XÎCh (A) is Gorenstein projective with respect to YX if and only if Xi is Gorenstein projective with respect to X for each i, when X is a self-orthogonal class or X is Hom (-, X) -exact. Subsequently, we consider the relationships of Gorenstein projective dimensions between them. As an application, if A is of finite left Gorenstein projective global dimension with respect to X and contains an in- jective cogenerator, then we find a new model structure on Ch (A) by Hovey’s results in [14].

Throughout this paper, (R, m) is a commutative Noetherian local ring with the maximal ideal m. The following conjecture proposed by Bass [1], has been proved by Peskin and Szpiro [2] for almost all rings: (B) If R admits a finitely generated R-module of finite injective dimension, then R is Cohen-Macaulay. The problems treated in this paper are closely related to the following generalization of Bass conjecture which is still wide open: (GB) If R admits a finitely generated R-module of finite Gorenstein-injective dimension, then R is Cohen-Macaulay. Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a finitely generated R-module of injective dimension ≤ 1. Levin and Vasconcelos indicate that if x m\m2 is a non-zerodivisor, then for every finitely generated R/xR-module M, there is id R M= id R/xr M+1. Using this fact, they construct a finitely generated R-module of finite injective dimension in the case where R is Cohen-Macaulay (the converse of Conjecture B). In this paper we study the Gorenstein injective dimension of local cohomology. We also show that if R is Cohen-Macaulay with minimal multiplicity, then every finitely generated module of finite Gorenstein injective dimension has finite injective dimension. We prove that a Cohen-Macaulay local ring has a finitely generated module of finite Gorenstein injective dimension.

We investigate the relative cohomology and relative homol-ogy theories of F -Gorenstein modules, consider the relations between classical and F -Gorenstein (co) homology theories.

Let H be a Hopf algebra and A an H-bimodule algebra. In this paper, we investigate Gorenstein global dimensions for Hopf algebras and twisted smash product algebras A * H. Results from the literature are generalized.