IFP-FLAT DIMENSIONS AND IFP-INJECTIVE DIMENSIONS

In basic homological algebra,the flat and injective dimensions of modules play an important and fundamental role.In this paper,the closely related I F P-flat and I F P-injective dimensions are introduced and studied.We show that I F P-fd(M) = I F P-id(M +) and I F P-fd(M +)=I F P-id(M) for any R-module M over any ring R.Let IIn(resp.,IFn) be the class of all left(resp.,right) R-modules of IFP-injective(resp.,I F P-flat) dimension at most n.We prove that every right R-module has an IF npreenvelope,(IF n,IF n⊥) is a perfect cotorsion theory over any ring R,and for any ring R with I F P-id(RR) ≤ n,(II n,II n⊥) is a perfect cotorsion theory.This generalizes and improves the earlier work(J.Algebra 242(2001),447-459).Finally,some applications are given.     

Simple-Baer Rings and Minannihilator Modules

Let R be a ring. M is said to be a minannihilator left R-module if r _M l _R (I) = IM for any simple right ideal I of R. A right R-module N is called simple-flat if Nl _R (I) = l _N (I) for any simple right ideal I of R. R is said to be a left simple-Baer (resp., left simple-coherent) ring if the left annihilator of every simple right ideal is a direct summand of _R R (resp., finitely generated). We first obtain some properties of minannihilator and simple-flat modules. Then we characterize simple-coherent rings, simple-Baer rings, and universally mininjective rings using minannihilator and simple-flat modules.     

Almost-Perfect Rings and Modules

Following [1 Amini , A. , Ershad , M. , Sharif , H. ( 2008 ). Rings over which flat covers of finitely generated modules are projective . Comm. Algebra 36 : 2862 – 2871 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.     

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.     

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.     

Notes on Divisible and Torsionfree Modules

In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied.     

Gorenstein Modules and Auslander Categories

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

7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.     

$PS$-Injective Modules, $PS$-Flat Modules and $PS$-Coherent Rings

A left ideal $I$ of a ring $R$ is small in case for every proper left ideal $K$ of $R, K +I≠R$. A ring $R$ is called left $PS$-coherent if every principally small left ideal $Ra$ is finitely presented. We develop, in this paper, $PS$-coherent rings as a generalization of $P$-coherent rings and $J$-coherent rings. To characterize $PS$-coherent rings, we first introduce $PS$-injective and $PS$-flat modules, and discuss the relation between them over some spacial rings. Some properties of left $PS$-coherent rings are also studied.     

Almost-Flat Modules

We present general properties for almost-flat modules and we prove that a self-small right module is almost flat as a left module over its endomorphism ring if and only if the class of g-static modules is closed under the kernels.     

Zip模(英文)

A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.     

Modules over serial rings

This paper continues the study of Noetherian serial rings. General theorems describing the structure of such rings are proved. In particular, some results concerning π-projective and π-injective modules over serial rings are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 880–892 June, 1999.     

Generalization of Regular Modules

We study generalizations of regular modules by Ramamurthy and Mabuchi. These are also generalizations of fully right idempotent and fully left idempotent rings, respectively. We also define and study the properties of *-weakly regular modules, a generalization of fully idempotent rings.     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

On ADS Modules and Rings

A right module M over a ring R is said to be ADS if for every decomposition M = S ⊕ T and every complement T′ of S, we have M = S ⊕ T′. In this article, we study and provide several new characterizations of this new class of modules. We prove that M is semisimple if and only if every module in σ[M] is ADS. SC and SI rings also characterized by the ADS notion. A ring R is right SC-ring if and only if every 2-generated singular R-module is ADS.