Strongly Gorenstein projective, injective and flat modules

Strongly Gorenstein projective, injective, and flat modules

Totally acyclic complexes

We characterize Ding modules and complexes over Ding-Chen rings. We show that, over a Ding-Chen ring $R$, the Ding projective (respectively, Ding injective, respectively, Ding flat) $R$-modules coincide with the Gorenstein projective (respectively, Gorenstein injective, respectively, Gorenstein flat) modules, which, in turn, are noth\-ing more than modules appearing as a cycle of an exact complex of projective (respectively, injective, respectively, flat) modules. We prove a similar characterization for chain complexes of $R$-modules: a complex~$X$ is Ding projective (respectively, Ding injective, respectively, Ding flat) if and only if each component $X_n$ is Ding projective (respectively, Ding injective, respectively, Ding flat). Along the way, we generalize some results of Stovicek and Bravo, Gillespie and Hovey to obtain other interesting corollaries. For example, we show that, over any Noetherian ring, any exact chain complex with Gorenstein injective components must have all cotorsion cycle modules, that is, $Ext ^1_R(F,Z_nI) = 0$ for any such complex $I$ and flat module $F$. On the other hand, over any coherent ring, the cycles of any exact complex $P$ with projective components must satisfy $Ext ^1_R(Z_nP,A) = 0$ for any absolutely pure module~$A$.

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.

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

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.

Let S and R be rings, be a semidualizing bimodule and be an integer or We introduce and study C-FP n -injective and C-FP n -flat modules as a common generalization of some known modules such as C-injective (resp. C-FP-injective, C-weak injective) and C-flat (resp. C-projective, C-weak flat) modules. Suppose that is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FP n -flat left S-modules and C-FP n -injective left R-modules. Then we show that the pairs and are coproduct-closed and product-closed duality pairs and both and are covering and preenveloping, where and denote the classes of C-FP n -flat left S-modules and C-FP n -injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FP n -injective and C-FP n -flat modules.

Gorenstein homological dimensions and Auslander categories

In this paper, we construct some non-integrally closed domains in Gorenstein multiplicative ideal theory. For example, we show that there exists a Gorenstein Prüfer domain which is neither Gorenstein Dedekind nor Prüfer, and there exists a Gorenstein Krull domain which is neither Gorenstein Dedekind nor Krull. Also, we construct a non-integrally closed non-coherent domain in which all Gorenstein projective (resp., injective, flat) modules are projective (resp., injective, flat).

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 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.

We introduce and investigate the notion of $\gc$-projective modules over (possibly non-noetherian) commutative rings, where $C$ is a semidualizing module. This extends Holm and J{\o}rgensen's notion of $C$-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite $\gc$-projective dimension, showing in particular that they admit $\gc$-projective approximations, a generalization of the maximal Cohen-Macaulay approximations of Auslander and Buchweitz. Over a local (noetherian) ring, we provide necessary and sufficient conditions for a $G_C$-approximation to be minimal.

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial . This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility. The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealiser subrings which forms the first part of the book and provides a useful general construction tool for interesting examples. The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

The main aim of this paper is to investigate rings over which all (finitely generated strongly) Gorenstein projective modules are projective.We consider this propriety under change of rings, and give various examples of rings with and without this propriety.

The early papers of Hyman Bass in the late 50s and the early 60s leading up to his pioneering work in algebraic K-theory have played an important and very special role in ring theory and the theory of projective (and injective) modules. In this article, we give a general survey of Bass's fundamental contributions in this early period of his work, and explain how much this work has influenced and shaped the thinking of subsequent researchers in the area.