A Generalization of Gorenstein Injective and Flat Modules

In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.

n-strongly Gorenstein Projective and Injective and Flat Modules

In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given.

Notes on Strongly n-Gorenstein Projective, Injective and Flat Modules

In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.

Remarks on Gorenstein Weak Injective and Weak Flat Modules

In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules.In this new setting,we characterize rings over which all modules are Gorenstein weak injective.Moreover,we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module,and also the stability of Gorenstein weak injective modules.

Strongly Gorenstein Flat Modules and Dimensions

There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.

Prime Submodules and Flat Modules

In this paper, some characterizations of prime submodules in flat modules and, particularly, in free modules are given. Furthermore, the height of prime submodules and some saturated chain of prime submodules are also given.

Ding w-Flat Modules and Dimensions

The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.

ABSOLUTELY E-PURE MODULES AND E-PURE SPLIT MODULES

We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.

On Stability of F-Gorenstein Flat Categories

To investigate cohomology theories based on flats, Asadollahi and Salarian gave the definition of F-Gorenstein flat R-modules, and these modules are exactly Gorenstein fiat provided that R is right coherent. In this paper, we get some properties of F-Gorenstein flat R-modules and establish the stability of F-Gorenstein flat categories.

Strongly Gorenstein Flat Dimensions

This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.

Completed Tensor Product and Flatness

We give,in a more general case than the Noetherian case,an answer to the question posed by Shaul on the flatness of the completed tensor product.

Generation of ultra-wide and flat optical frequency comb based on electro-absorption modulator and frequency modulator

An ultra-wide and flat optical frequency comb(OFC) generation scheme using multiple continuous wave(CW) light sources based on electro-absorption modulator(EAM) and frequency modulator(FM) is proposed. In the scheme, each CW light source is broadened and modulated by the first EAM and FM, respectively. The second EAM is introduced to flatten the ultra-wide OFC lines. By setting the wavelength spacing of light sources equal to the bandwidth of sub-OFC, an ultra-wide OFC can be obtained. Principle analysis and simulation for the scheme are performed. The results show that in the case of a single light source, a tunable and flat OFC is obtained. With the increase of light sources, the bandwidth of the generated ultra-wide OFC expands rapidly. In the case of 28 light sources, a 22 GHz ultra-wide OFC with bandwidth of 16.52 THz can be generated.

Acyclic Complexes and Gorenstein Rings

For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that the following are equivalent over any ring R:every exact complex of injective modules is totally acyclic;every exact complex of Gorenstein injective modules is in every complex in dw(GL)is dg-Gorenstein injective.The analogous result for complexes of flat and Gorenstein flat modules also holds over arb计rary rings.If the ring is n-perfect for some integer n≥0,the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules.We also prove the following characterization of Gorenstein rings.Let R be a commutative coherent ring;then the following are equivalent:(1)every exact complex of FP-injective modules has all its cycles Ding injective modules;(2)every exact complex of flat modules is F-totally acyclic,and every R-modulc M such that M^(+)is Gorenstein flat is Ding injective;(3)every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that is Gorenstein flat is Ding injective.If R has finite Krull dimension,statements(1)-(3)are equivalent to(4)R is a Gorenstein ring(in the sense of Iwanaga).

Some Criteria for the Cohen-Macaulay Property and Local Cohomology

Let a be an ideal of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d. We characterize Cohen-Macaulay rings in term of a special homological dimension. Lastly, we prove that if R is a complete local ring, then the Matlis dual of top local cohomology module Ha^d（M） is a Cohen-Macaulay R-module provided that the R-module M satisfies some conditions.

A simplified tunable frequency interval optical frequency comb generator using a single continuous-wave laser

We propose and demonstrate a simplified and tunable frequency interval optical frequency comb(OFC) generator based on a dual-drive Mach-Zehnder modulator(DD-MZM) using a single continuous-wave(CW) laser and low-power radio frequency(RF) driven signal. A mathematical model for the scheme is established. The 21-and 29-mode OFCs with frequency interval ranging from 6 GHz to 40 GHz are obtained under DD-MZM driven by a low-power RF signal within a maximum bandwidth of 1.12 THz. The generated OFCs exhibit spectral flatnesses of less than 0.5 d B and 0.8 d B within bandwidths of 160 GHz and 400 GHz, respectively.