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The main purpose of this paper is to generalize a result of T.G. Lucas which states that if R is a reduced commutative ring and M is a flat R-module, then the idealization R ⋉ M is an A -ring if and only if R is an A -ring [14, Proposition 3.5]. In effect, we drop the reduceness hypotheses and prove that, given an arbitrary commutative ring R and any submodule M of a flat R-module F, R ⋉ M is an A -ring (resp., S A -ring) if and only if R is an A -ring (resp., S A -ring). [ABSTRACT FROM AUTHOR]

Several authors have been interested in cotorsion theories. Among these theories we figure the pairs (Pn, Pn ⊥), where Pn designates the set of modules of projective dimension at most a given integer n > 1 over a ring R. In this paper, we shall focus on homological properties of the class P-1 ⊥ that we term the class of P1-injective modules. Numerous nice characterizations of rings as well as of their homological dimensions arise from this study. In particular, it is shown that a ring R is left hereditary if and only if any P1-injective module is injective and that R is left semi-hereditary if and only if any P1-injective module is FP-injective. Moreover, we prove that the global dimensions of R might be computed in terms of P1-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows wgl-dim(R) = sup{fdR(M) : M is a P1-injective left R-module} and l-gl-dim(R) = sup{pdR(M) : M is a P1-injective left R-module}. We close the paper by proving that, given a Matlis domain R and an R-module M ∈ P1, HomR(M, N) is P1-injective for each P1-injective module N if and only if M is strongly flat. [ABSTRACT FROM AUTHOR]

The theory of Gorenstein flat dimension is not complete since it is not yet known whether the category 풢ℱ(R) of Gorenstein flat modules over a ring R is projectively resolving or not. Besides, it arises from recent investigations on this subject that there exists several ways of measuring the Gorenstein flat dimension of modules which turn out to coincide with the usual one in the case where 풢ℱ(R) is projectively resolving. These alternate procedures yield new invariants which enjoy very nice behavior for an arbitrary ring R. In this paper, we introduce and study one of these invariants called the cover Gorenstein flat dimension of a module M and denoted by R(M). This new entity stems from a sort of a Gorenstein flat precover of M. First, for each R-module M, we prove that R(M) ≤ R(M) for each R-module M with whenever R(M) is finite. Also, we show that 풢ℱ(R) is projectively resolving if and only if the Gorenstein flat dimension and the introduced cover Gorenstein flat dimension coincide. In particular, if R is a right coherent ring, then R(M) = R(M) for any R-module M. As a consequence, we prove that if R is a left and right GF-closed, then the Gorenstein weak global dimension of R is left-right symmetric and it is related to the cohomological invariants leftsfli(R) and rightsfli(R) by the formula [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex R(H, G) is exact for each Gorenstein injective R-module H, is the module (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring. [ABSTRACT FROM AUTHOR]

This paper is concerned with the second change of rings theorems for the Gorenstein homological dimensions. More precisely, we give the ultimate versions of these theorems relatively to these new invariants. [ABSTRACT FROM AUTHOR]

Sather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module? [ABSTRACT FROM PUBLISHER]

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45-67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules. [ABSTRACT FROM AUTHOR]