Some Concepts Related to Almost Injective(Projective)Semi modules

Totally acyclic complexes

Strongly Gorenstein projective, injective, and flat modules

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.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

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.

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.

Strongly Gorenstein projective, injective and flat modules

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.

Projective modules: Modules and homomorphisms Projective modules Completely reducible modules Wedderbum rings Artinian rings Hereditary rings Dedekind domains Projective dimension Tensor products Local rings Polynomial rings: Skew polynomial rings Grothendieck groups Graded rings and modules Induced modules Syzygy theorem Patching theorem Serre conjecture Big projectives Generic flatness Nullstellensatz Injective modules: Injective modules Injective dimension Essential extensions Maximal ring of quotients Classical ring of quotients Goldie rings Uniform dimension Uniform injective modules Reduced rank Index.

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.

The present chapter is devoted to the study of Noetherian rings, Projective modules, Injective Modules, Tensor product of modules, Grothendieck, and Whitehead groups of rings.

In this paper, the primary objective is to obtain decomposition theorems for graded modules over the polynomial ring $k[x]$, where $k$ denotes a field. There is some overlap with recent work of Höppner and Lenzing. The results obtained include identification of the free, projective, and injective modules. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules. Pure submodules are direct summands if they are bounded below. In such case, the pure submodule is itself a direct sum of cyclic submodules. It is also noted that Cohen and Gluck’s Stacked Bases Theorem remains true if the modules are graded.

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules.

Does Hopkins' theorem extend to projective modules, i.e., are projective Artinian modules Noetherian?An example is given to answer this question in the negative; however, we show that the answer is affirmative for certain large classes of projective modules.Dually, are injective Noetherian modules Artinian?Again the answer is negative; nevertheless, we provide an affirmativeâ nswer for certain classes of injective modules.

The focus of this book is the study of the noncommutative aspects of rings and modules, and the style will make it accessible to anyone with a background in basic abstract algebra. Features of interest include an early introduction of projective and injective modules; a module theoretic approach to the Jacobson radical and the Artin-Wedderburn theorem; the use of Baer's criterion for injectivity to prove the structure theorem for finitely generated modules over a principal ideal domain; and applications of the general theory to the representation theory of finite groups. Optional material includes a section on modules over the Weyl algebras and a section on Goldie's theorem. When compared to other more encyclopedic texts, the sharp focus of this book accommodates students meeting this material for the first time. It can be used as a first-year graduate text or as a reference for advanced undergraduates.