Weak Global Dimension of Coherent Rings

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ? ? the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ? ?-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n ? 2)th ? ?-syzygy in a minimal ? ?-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Strongly FP-injective modules

A module M is called strongly FP-injective if Extⁱ(P,M) = 0 for any finitely presented module P and all i≥1. (Pre)envelopes and (pre)covers by strongly FP-injective modules are studied. We also use these modules to characterize coherent rings. An example is given to show that (strongly) FP-injective (pre)covers may fail to be exist in general. We also give an example of a module that is FP-injective but not strongly FP-injective.     

On Strongly Copure Flat Modules and Copure Flat Dimensions

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext ⁱ (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings.     

Relative <Emphasis Type="Italic">FI</Emphasis>-injective and <Emphasis Type="Italic">FI</Emphasis>-flat modules

Let R be a ring, n a fixed nonnegative integer and FP _n (F _n ) the class of all left (right) R-modules of FP-injective (flat) dimensions at most n. A left R-module M (resp., right R-module F) is called n-FI-injective (resp., n-FI-flat) if Ext ¹(N,M) = 0 (resp., Tor ₁(F,N) = 0) for any N ∈ FP _n . It is shown that a left R-module M over any ring R is n-FI-injective if and only if M is a kernel of an FP _n -precover f: A → B with A injective. For a left coherent ring R, it is proven that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an F _n -preenvelope K → F of a right R-module K with F projective if and only if M ∈^⊥ F _n . These classes of modules are used to construct cotorsion theories and to characterize the global dimension of a ring.     

On Divisible and Torsionfree Modules

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext¹(G, M) = 0 (resp. Tor₁(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.     

On Covers and Envelopes in Some Functor Categories

We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every left R-module M has an Ext-phantom preenvelope f: M → N with coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by the FP-injective objects.     

Symplectic algebras and poisson algebras

Abstract

A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l( ? ) and r( ? ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 _n (R) is left GP-injective.     

On simple singular gp - injective modules

We investigate von Neumann regularity of rings whose simple singular right R-modules are GP-injective. It is proved that a ring; R is strongly regular iff R is a weakly right duo ring whose simple singular right R-modules are GP-injective. And it is also shown that R is either a strongly right bounded ring or a zero insertive ring in which every simple singular right R-module is GP-injective are reduced weakly regular rings. Several known results are unified and extended.     

FP-Injective and Weakly Quasi-Frobenius Rings

The classes of FP-injective and weakly quasi-Frobenius rings are investigated. The properties of both classes of rings are closely related to the embedding of finitely presented modules in fp-flat and free modules, respectively. Using these properties, we characterize the classes of coherent CF- and FGF-rings. Moreover, it is proved that the group ring R(G) is FP-injective (weakly quasi-Frobenius, respectively) if and only if the ring R is FP-injective (weakly quasi-Frobenius) and G is locally finite. Bibliography: 15 titles.     

When Cyclic Modules Have Σ-Injective Hulls

Abstract

A theorem of Cartan-Eilenberg (Cartan, H., Eilenberg, S. (1956). Homological Algebra. Princeton: Princeton University Press, pp. 390.) states that a ring Ris right Noetherian iff every injective right module is Σ-incentive. The purpose of this paper is to study rings with the property, called right CSI, that, all cyclic right R-modules have Σ-injective hulls, i.e., injective hulls of cyclic right R-modules are Σ-injective. In this case, all finitely generated right R-modules have Σ-injective hulls, and this implies that Ris right Noetherian for a lengthy list of rings, most notably, for Rcommutative, or when Rhas at most finitely many simple right R-modules, e.g., when Ris semilocal. Whether all right CSIrings are Noetherian is an open question. However, if in addition, R/rad Ris either right Kasch or von Neuman regular (=VNR), or if all countably generated (sermisimple) right R-modules have Σ-injective hulls then the answer is affirmative. (See Theorem A.) We also prove the dual theorems for Δ-injective modules.     

IFP-Flat Modules and IFP-Injective Modules

In this article, we introduce the concept of IFP-flat (resp., IFP-injective) modules as nontrivial generalization of flat (resp., injective) modules. We investigate the properties of these modules in various ways. For example, we show that the class of IFP-flat (resp., IFP-injective) modules is closed under direct products and direct sums. Therefore, the direct product of flat modules is not flat in general; however, the direct product of flat modules is IFP-flat over any ring. We prove that (^⊥??, ??) is a complete cotorsion theory and (??, ??^⊥) is a perfect cotorsion theory, where ?? stands for the class of all IFP-injective left R-modules, and ?? denotes the class of all IFP-flat right R-modules.     

Gorenstein FP-Injective Dimension for Complexes

This article is a continuous work of [17 Hu , J. , Zhang , D. ( 2013 ). Weak AB-context for FP-injective modules with respect to semidualizing modules . J. Algebra Appl. 12 ( 7 ): 1350039 .[Crossref], [Web of Science ®] , [Google Scholar]], where the coauthors introduced the notion of 𝒢-FP-injective R-modules. In this article, we define a notion of 𝒢-FP-injective dimension for complexes over left coherent rings. To investigate the relationships between 𝒢-FP-injective dimension and FP-injective dimension for complexes, the complete cohomology group bases on FP-injectives is given.     

Relative Projective Modules and Relative Injective Modules

Let R be a ring, and n and d fixed non-negative integers. An R-module M is called (n, d)-injective if Ext ^d+1 _R (P, M) = 0 for any n-presented R-module P. M is said to be (n, d)-projective if Ext¹ _R (M, N) = 0 for any (n, d)-injective R-module N. We use these concepts to characterize n-coherent rings and (n, d)-rings. Some known results are extended.     

An analogue of nakai's conjecture

Let R be an associative ring with 1 and let T be a hereditary torsion theory in the category of left R-modules. In defining the localizatio n of R respect to x, the concept of T-injective module arises (see [5] , [11]) . We can consider the family E _T of all short exact sequences of left R-modules respect to which every T-injective left R-module is injective . E _T proper class in the sense defined in [ 9] . In this paper we characterize proper classe s which are of the form E _T for some hereditary torsion theory x. On the other hand, we give some conditions on x, which imply that E _T has enough projectives , and we show an example where E _T does not have enough projectives.     

Characterizations of QF Rings

A ring R is called “left GP-injective” if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). It is proved that (1) every right Noetherian left GP-injective ring such that every complement left ideal is a left annihilator is a QF ring, (2) every left GP-injective ring with ACC on left annihilators such that every complement left ideal is a left annihilator is a QF ring, and (3) every left P-injective left CS ring satisfying ACC on essential right ideals is a QF ring. Several well-known results on QF rings are obtained as corollaries.     

Comparison of Some Purities,Flatnesses and Injectivities

In this article, we compare (n, m)-purities for different pairs of positive integers (n, m). When R is a commutative ring, these purities are not equivalent if R does not satisfy the following property: there exists a positive integer p such that, for each maximal ideal P, every finitely generated ideal of R _P is p-generated. When this property holds, then the (n, m)-purity and the (n, m′)-purity are equivalent if m and m′ are integers ≥np. These results are obtained by a generalization of Warfield's methods. There are also some interesting results when R is a semiperfect strongly π-regular ring. We also compare (n, m)-flatnesses and (n, m)-injectivities for different pairs of positive integers (n, m). In particular, if R is right perfect and right self (?₀, 1)-injective, then each (1, 1)-flat right R-module is projective. In several cases, for each positive integer p, all (n, p)-flatnesses are equivalent. But there are some examples where the (1, p)-flatness is not equivalent to the (1, p + 1)-flatness.     

(n,d)-Injective covers,n-coherent rings,and (n,d)-rings

It is known that a ring R is left Noetherian if and only if every left R-module has an injective (pre)cover. We show that (1) if R is a right n-coherent ring, then every right R-module has an (n, d)-injective (pre)cover; (2) if R is a ring such that every (n, 0)-injective right R-module is n-pure extending, and if every right R-module has an (n, 0)-injective cover, then R is right n-coherent. As applications of these results, we give some characterizations of (n, d)-rings, von Neumann regular rings and semisimple rings.     

On a Generalization of Injective Rings

Abstract

A ring R is called left IP-injective if every homomorphism from a left ideal of R into R with principal image is given by right multiplication by an element of R. It is shown that R is left IP-injective if and only if R is left P-injective and left GIN (i.e., r(I ∩ K) = r(I) + r(K) for each pair of left ideals I and K of R with I principal). We prove that R is QF if and only if R is right noetherian and left IP-injective if and only if R is left perfect, left GIN and right simple-injective. We also show that, for a right CF left GIN-ring R, R is QF if and only if Soc(R _R ) ? Soc( _R R). Two examples are given to show that an IP-injective ring need not be self-injective and a right IP-injective ring is not necessarily left IP-injective respectively.