The Existence of Flat Covers

Totally acyclic complexes

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 ...

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.

Abstract.Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module RU to have an indecomposable local dual

Let R be a commutative ring and C a semidualizing R-module. We investigate the relations between C-flat modules and C-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

A right R-module M is said to be finendo provided that as a left module over its endomorphism ring it is finitely generated. Let B = End MR. Then Bn → M → 0 is exact for some finite integer n. This implies (and when M is quasi-injective is implied by) the exactitude of 0 → R/annRM → Mn. This implies that a finendo module is quasi iff injective modulo annihilator (1). Similarly for any finitely generated module over a commutative ring. Any finendo quasi-injective module is injective over its biendomorphism ring (17B). A ring R is right Artinian iff every quasi-injective right module is finendo (17). This supplies the converse of a theorem of K. Fuller [69]. Moreover, a ring R is right self-injective iff every finendo faithful (injective) right R-module generates mod-R (18B). Azumaya defined a right PF ring as one such that every faithful right R-module generates mod-R. We characterize a right PF ring as a right self-injective ring such that every faithful (quasi) injective right R-module is finendo (38). Then, if every factor ring is PF, the ring must be right Artinian, hence uniserial (40). In general, a factor ring R/A of a right PF ring is right PF iff the left annihilator of A is Rz = zR for some z e R (39).

Given a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.

ABSTRACTFor a (right and left) coherent ring A, we show that there exists a duality between homotopy categories and 𝕂b(mod-A). If A = Λ is an artin algebra of finite global dimension, this duality induces a duality between their subcategories of acyclic complexes, and As a result, it will be shown that, in this case, admits a Serre functor and hence has Auslander–Reiten triangles.

In this paper we generalize the notion of pure injectivity of modules by introducing what we call a pure Baer injective module. Some properties and some characterization of such modules are established. We also introduce two notions closely related to pure Baer injectivity; namely, the notions of a ∑‐pure Baer injective module and that of SSBI‐ring. A ring R is an SSBI‐ring if and only if every smisimple R‐module is pure Baer injective. To investigate such algebraic structures we had to define what we call p‐essential extension modules, pure relative complement submodules, left pure hereditary rings and some other related notions. The basic properties of these concepts and their interrelationships are explored, and are further related to the notions of pure split modules.

Pure injective modules over hereditary Noetherian prime rings with enough invertible ideals

One of the main results of this paper is the characterization of left FP-injective rings as rings over which every finitely presented right module is torsionless. The necessity of the torsionless condition (in fact reflexivity) previously has been observed for special classes of FP-injective rings (e.g., self-injective rings (Kato and McRae), right and left coherent and self FP-injective rings (Stenström)). Furthermore, if the ring is left coherent, it is equivalent to say that every finitely presented right module is reflexive. We investigate the condition (called IF) that every injective right module is flat and characterize this via FP-injectivity. IF-rings generalize regular rings and QF-rings, and under certain chain or homological conditions coincide with regular or QF-rings (e.g., w g l dim ⁡ R ≦ 1 ⇒ regular wgl\dim R \leqq 1 \Rightarrow {\text {regular}} , noetherian, or right perfect ⇒ QF) {\text {right perfect}} \Rightarrow {\text {QF)}} .

It is proved that an injective module after a flat change of base is of pointwise finite injective dimension if and only if all the fibers at points associated to the injective module are Gorenstein rings (or trivial).

We work over coherent rings. We consider the class of weakly Ding injective modules. These are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor Hom ( A , − ) for any FP-injective module A. Throughout the paper, we assume that the class of weakly Ding injective modules, w D I , is closed under extensions. We prove that any weakly Ding injective module is a direct sum of the form F ⊕ D with F FP-injective and with D Ding injective. We also prove that the class of weakly Ding injective modules is the right half of a complete hereditary cotorsion pair. Then we show that w D I is ccovering if and only if it is closed under direct limits.