FP n -injective and FP n -flat modules with respect to a semidualizing bimodule

Totally acyclic complexes

Strongly Gorenstein projective, injective, and flat modules

Let R be a ring and n be a positive integer. R is called left n-coherent, if every n-presented left R-module is $$(n+1)$$ -presented. A left R-module M is called (n, 0)-injective if Ext $$^1_R(V, M)=0$$ for every n-presented module V, a right R-module F is called (n, 0)-flat if Tor $$^R_1(F, V)=0$$ for every n-presented module V, a left R-module M is called (n, 0)-projective if $$\mathrm{Ext}^1_R(M, N)=0$$ for any (n, 0)-injective module N, and a right R-module M is called (n, 0)-cotorsion if $$\mathrm{Ext}^1_R(F, M)=0$$ for any (n, 0)-flat module F. We give some characterizations and properties of (n, 0)-projective modules and (n, 0)-cotorsion modules. n-coherent rings, n-hereditary rings and n-regular rings are characterized by (n, 0)-projective modules, (n, 0)-cotorsion modules, (n, 0)-injective modules and (n, 0)-flat modules.

Strongly Gorenstein projective, injective and flat modules

In this paper, we introduce Gorenstein weak injective and weak flat modules in terms of, respectively, weak injective and weak flat modules; the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules. In this new setting, we characterize rings over which all modules are Gorenstein weak injective. Moreover, we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module, and also the stability of Gorenstein weak injective modules.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if $$Ext^{1}_{R}(F/K, A) = 0$$ (resp. $${\text {Tor}}_{1}^{R}(B, F/K)= 0$$ ) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.

On the dual of a flat module in TOP

A left R-module M is called almost F-injective, if every R-homomorphism from a finitely presented left ideal to M extends to a homomorphism of R to M. A right R-module V is said to be almost flat, if for every finitely presented left ideal I, the canonical map V ⊗ I → V ⊗R is monic. A ring R is called left almost semihereditary, if every finitely presented left ideal of R is projective. A ring R is said to be left almost regular, if every finitely presented left ideal of R is a direct summand of RR. We observe some characterizations and properties of almost F-injective modules and almost flat modules. Using the concepts of almost F-injectivity and almost flatness of modules, we present some characterizations of left coherent rings, left almost semihereditary rings, and left almost regular rings.

This is an introductory text on the theory of modules over rings or, equivalently, over algebras, addressed to beginning graduate students. It stresses the importance of the categorical perspective and the use of homological tools. Modules are studied both from a classical point of view and a categorical one. For this purpose, rudiments of category theory, homological algebra and representations of quivers are introduced and applied in the study of module categories. After introducing such fundamental tools as the Hom functor and the tensor product comes the study of projective, injective and flat modules. The classical results, such as the classification of modules over principal ideal domains, the Jordan-Hölder, Wedderburn-Artin, and Morita theorems, are proven, as well as the Eilenberg-Watts theorems. Radicals (of modules, algebras and categories) are thoroughly studied. Homological invariants, such as the extension and torsion modules or the homological dimensions are introduced using derived functors and are applied in module theory. The book stresses the construction and analysis of examples.

Flat endofinite modules, prime ideals, and duality

In this article, some new characterizations of Gorenstein projective, injective, and flat modules over commutative noetherian local rings are given.

Minimal pure injective resolutions of flat modules

In this paper we characterize the relative Gorenstein weak global dimension of the Gorenstein $$\mathcal {B}$$ B -flat R-modules and projectively coresolved Gorenstein $$\mathcal {B}$$ B -flat R-modules recently studied by S. Estrada, A. Iacob, and M. A. Pérez, which are a relativisation of the ones introduced by J. Šaroch and J. Št’ovíchěk. As application we prove that the weak global dimension with respect to the Gorenstein $$\textrm{FP}_n$$ FP n -flat R-modules is finite over a Gorenstein n-coherent ring R and in this case coincides with the flat dimension of the right $$\textrm{FP}_n$$ FP n -injective R-modules. This result extends the known for Gorenstein flat modules over Iwanaga-Gorenstein and Ding-Chen rings. We also show that there is a close relationship between the relative global dimension of the Gorenstein $$\textrm{FP}_n$$ FP n -projectives and the Gorenstein weak global dimension respect to the class of Gorenstein $$\textrm{FP}_n$$ FP n -flat R-modules. We also get an hereditary and complete cotorsion triple and consequently a balanced pair.

Over any associative ring R R it is standard to derive H o m R ( − , − ) \mathrm {Hom}_R(-,-) using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains E x t R n ( − , − ) \mathrm {Ext}_R^n(-,-) in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product − ⊗ R − -\otimes _R- using Gorenstein flat modules.