‎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...

Let $R$ be a commutative Noetherian ring. We prove that  over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.

Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the...

This paper is a continuation of the papers J. Pure Appl. Algebra, 210 (2007), 437–445 and J. Algebra Appl., 8 (2009), 219–227. Namely, we introduce and study a doubly filtered set of classes of modules of finite Gorenstein projective dimension, which are called (n, m)-strongly Gorenstein projective ((n, m)-SG-projective for short) for integers n ≥ 1 and m ≥ 0. We are mainly interested in studyi...

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.

In 1966 [1], Auslander introduced a class of finitely generated modules having a certain complete resolution by projective modules. Then using these modules, he defined the G-dimension (G ostensibly for Gorenstein) of finitely generated modules. It seems appropriate then to call the modules of G-dimension 0 the Gorenstein projective modules. In [4], Gorenstein projective modules (whether finite...

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...

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.

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...

The Gorenstein projective modules are proved to form a precovering class in the module category of a ring which has a dualizing complex. 0. Introduction This paper proves over a wide class of rings that the Gorenstein projective modules form a precovering class in the module category. Let me explain this statement. There are two terms of mystery, “Gorenstein projective modules” and “precovering...