Nanqing Ding

The flat dimensions of injective modules

Let R be a ring. Define r. IFD(R) as r.IFD(R)=sup{fdE/E is an injective right R—module}. The purpose of this paper is to investigate this “global” dimension.

Coherent rings with finite self-FP-injective dimension

In this paper, we generalize the characterization of Gorenstein flat modules over Gorenstein rings to n − FC rings (coherent rings with finite sdf−FP−injective dimension), and characterize n − FC rings in terms of Gorenstein flat and projective modules.

On envelopes with the unique mapping property

We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) = 2) if and only if every (n – 2)th F–cosyzygy of a finitely presented right R–module has a flat envelope with the unique mapping property; (b) if R is a left coherent and right perfect ring, then the right global dimension rD(R) = 2) if and only if every (n – 2)th P–cosyzygy of a right R–module has a projective envelope with the unique mapping property; (c) if R is a commutative ring, then R is π—coherent (resp. coherent) and the exactness of 0 -> K -> F0 -> F1 with Fo and F1 (finitely) projective and K finitely generated implies the projectivity of K if and only if every finitely generated (resp, finitely presented) R–module has a (finitely) projective envelope with the unique mapping property.

GORENSTEIN FP-INJECTIVE AND GORENSTEIN FLAT MODULES

In this paper, Gorenstein FP-injective modules are introduced and studied. An R-module M is called Gorenstein FP-injective if there is an exact sequence ⋯ → E1 → E0 → E0 → E1 → ⋯ of injective R-modules with M = ker(E0 → E1) and such that Hom(E, -) leaves the sequence exact whenever E is an FP-injective R-module. Some properties of Gorenstein FP-injective and Gorenstein flat modules over coherent rings are obtained. Several known results are extended.

Relative coherence and preenvelopes

We define coherence relative to an arbitrary torsion theory and characterize it in terms of preenvelopes of modules. In particular, some known results are obtained as corollaries.

Envelopes and Covers by Modules of Finite FP-Injective and Flat Dimensions

Let R be a ring, n a fixed non-negative integer and ℱ ℐ n (ℱ n ) the class of all right (left) R-modules of FP-injective (flat) dimension at most n. We prove that ( is a perfect cotorsion theory if R is a right coherent ring with FP-id(R R ) ≤ n. This result was proven by Aldrich, Enochs, Jenda, and Oyonarte in Noetherian case. The modules in are also studied. Some applications are given.

On general principally injective rings

The aim of this paper is to investigate general principally injective rings satisfying additional conditions. Various results are developed, many extending known results.

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 (m, n)-INJECTIVITY OF MODULES

Let R be a ring. For two fixed positive integers m and n, a right R-module M is called (m, n)-injective if every right R-homomorphism from an n-generated submodule of Rm to M extends to one from Rm to M. This definition unifies several definitions on generalizations of injectivity of modules. The aim of this paper is to investigate properties of the (m, n)-injective modules. Various results are developed, many extending known results.

FP-PROJECTIVE DIMENSIONS

ABSTRACT We define a dimension, called an FP-projective dimension, for modules and rings. It measures how far away a finitely generated module is from being finitely presented, and how far away a ring is from being Noetherian. This dimension has nice properties when the ring in question is coherent. The relations between the FP-projective dimension and other homological dimensions are discussed.