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 applicati...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.展开更多
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, poly...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.展开更多
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 invest...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.展开更多
In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat an...In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat and Gorenstein FP-injective dimensions, and show that some characterizations similar to Gorenstein homological dimensions hold for these two dimensions.展开更多
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...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.展开更多
We consider the conditions under which the class of （m, d）-injective R-modules is （pre）covering. It is shown that every left R-module over a left （m, d）-coherent ring has an （rn, d）-injective cover. Moreover, ...We consider the conditions under which the class of （m, d）-injective R-modules is （pre）covering. It is shown that every left R-module over a left （m, d）-coherent ring has an （rn, d）-injective cover. Moreover, the classes of Gorenstein （m, d）-flat modules and Gorenstein （m, d）-injecitve modules are introduced and studied. For a right （m, d）-coherent ring R, we prove that a left R-module M is Gorenstein （m, d）-flat if and only if M＋ is Gorenstein （m, d）- injective as a right R-module. Some results on Gorenstein flat modules and Gorenstein n-flat modules are generalized.展开更多
In this article,we first establish a recollement related to projectively coresolved Gorenstein flat(PGF)complexes.Secondly,we define and study PGF dimension of complexes,we denote it PG F(X)for a complex X.It is shown...In this article,we first establish a recollement related to projectively coresolved Gorenstein flat(PGF)complexes.Secondly,we define and study PGF dimension of complexes,we denote it PG F(X)for a complex X.It is shown that the PGF(X)is equal to the infimum of the set{supA|there exists a diagram of morphisms of complexes A←G→X,such that G→X is a special PGF precover of X and G→A is a PGF almost isomorphism}.展开更多
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.
A novel technique is proposed for the generation of ultra-wide and flat optical frequency comb(OFC)based on serially cascading three stages of electro absorption modulators(EAMs)through sinusoidal radio frequency(RF)s...A novel technique is proposed for the generation of ultra-wide and flat optical frequency comb(OFC)based on serially cascading three stages of electro absorption modulators(EAMs)through sinusoidal radio frequency(RF)signals by setting frequencies at f GHz,f/2 GHz and f/4 GHz.Here,the first stage acts as subcarrier generator,the second stage acts as subcarrier doubler,and the third stage acts as subcarrier quadrupler.In addition,a higher number of subcarriers can easily be generated by adjusting the driving sinusoidal RF signal.In this paper,cascading three stages of EAMs driven by50 GHz,25 GHz and 12.5 GHz clock sources,we obtain 272 subcarriers with spacing of 2.5 GHz and power deviation within 1 d B.Theoretical analysis of serially cascaded EAMs for subcarrier generation is also investigated.Principal analysis and simulation of this technique are demonstrated.展开更多
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, ...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.展开更多
In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. W...In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd（M） = IFP-id（M＋） and IFP-fd（M＋）=IFP-id（M） for any R-module M over any ring R. Let ：Z-In （resp., ＂Zgv,~） he the class of all left （resp., right） R-modules of IFP-injective （resp., IFP-flat） dimension at most n. We prove that every right R-module has an IFn- preenvelope, （IFn,IF⊥n） is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id（RR） 〈 n, （IIn,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.展开更多
In this paper, a flatly broadened and highly coherent supercontinuum generation by induced modulation instability in ANDi-PCF is proposed. The numerical results show that the bandwidth of supercontinuun can be increas...In this paper, a flatly broadened and highly coherent supercontinuum generation by induced modulation instability in ANDi-PCF is proposed. The numerical results show that the bandwidth of supercontinuun can be increased by increasing the modulation depth and the coherence property of supercontinuum can be improved with the optimal modulation frequency. A highly coherent supercontinuum with the 10 dB bandwidth of 1305 nm and the flatness of less than 10 dB can be obtained when seeding at the modulation depth of 0.3 and modulation frequency of 24 THz.展开更多
LetRbeanarbitrary ring and let S be a separable and quasi-Frobenius extension of R.Then for any right S-module M,the Gorenstein flat dimensions of MS andMR areidentical.As a consequence,the Gorenstein weak global dime...LetRbeanarbitrary ring and let S be a separable and quasi-Frobenius extension of R.Then for any right S-module M,the Gorenstein flat dimensions of MS andMR areidentical.As a consequence,the Gorenstein weak global dimension of S is less than or equal to that of R.展开更多
Inspired by Bouchiba’s work on generalized Gorenstein flat modules,we introduce the generalized(F,A)-Gorenstein flat modules in this paper.By defining generalizedA-complete flat resolution,a new measure of the dimens...Inspired by Bouchiba’s work on generalized Gorenstein flat modules,we introduce the generalized(F,A)-Gorenstein flat modules in this paper.By defining generalizedA-complete flat resolution,a new measure of the dimension of generalized(F,A)-Gorenstein flat modules is given.When(⊥1,)AAis a complete cotorsion pair,R is a right coherent ring andAcontains injective right R-modules,it is proved that for each M with finite flat dimension,the generalized(F,A)-Gorenstein flat dimension of M is equal to the(F,A)-Gorenstein flat dimension of M.展开更多
基金Supported by the National Natural Science Foundation of China（11361051） Supported by the Program for New Century Excellent the Talents in University（NCET-13-0957）
文摘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.
基金Supported by the NNSF of China(10901129)Supported by the SRFDP(20096203120001)
文摘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.
基金The NSF(11501451)of Chinathe Fundamental Research Funds(31920150038)for the Central Universities and XBMUYJRC(201406)
文摘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.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1120137711261050)+1 种基金China Postdoctoral Science Foundation(Grant No.2013M541509)Program of Science and Technique of Gansu Province(Grant No.1208RJZA145)
文摘In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat and Gorenstein FP-injective dimensions, and show that some characterizations similar to Gorenstein homological dimensions hold for these two dimensions.
文摘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.
基金Supported by the Provincial Natural Science Research Program of Higher Education Institution of Anhui Province(Grant No.KJ2012Z028)
文摘We consider the conditions under which the class of （m, d）-injective R-modules is （pre）covering. It is shown that every left R-module over a left （m, d）-coherent ring has an （rn, d）-injective cover. Moreover, the classes of Gorenstein （m, d）-flat modules and Gorenstein （m, d）-injecitve modules are introduced and studied. For a right （m, d）-coherent ring R, we prove that a left R-module M is Gorenstein （m, d）-flat if and only if M＋ is Gorenstein （m, d）- injective as a right R-module. Some results on Gorenstein flat modules and Gorenstein n-flat modules are generalized.
基金Supported by the National Natural Science Foundation of China(12061061)Young Talents Team Project of Gansu Province(2025QNTD49)+1 种基金Lanshan Talents Project of Northwest Minzu University(Xbmulsrc202412)Longyuan Young Talents of Gansu Province。
文摘In this article,we first establish a recollement related to projectively coresolved Gorenstein flat(PGF)complexes.Secondly,we define and study PGF dimension of complexes,we denote it PG F(X)for a complex X.It is shown that the PGF(X)is equal to the infimum of the set{supA|there exists a diagram of morphisms of complexes A←G→X,such that G→X is a special PGF precover of X and G→A is a PGF almost isomorphism}.
文摘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.
文摘A novel technique is proposed for the generation of ultra-wide and flat optical frequency comb(OFC)based on serially cascading three stages of electro absorption modulators(EAMs)through sinusoidal radio frequency(RF)signals by setting frequencies at f GHz,f/2 GHz and f/4 GHz.Here,the first stage acts as subcarrier generator,the second stage acts as subcarrier doubler,and the third stage acts as subcarrier quadrupler.In addition,a higher number of subcarriers can easily be generated by adjusting the driving sinusoidal RF signal.In this paper,cascading three stages of EAMs driven by50 GHz,25 GHz and 12.5 GHz clock sources,we obtain 272 subcarriers with spacing of 2.5 GHz and power deviation within 1 d B.Theoretical analysis of serially cascaded EAMs for subcarrier generation is also investigated.Principal analysis and simulation of this technique are demonstrated.
基金supported by the Specialized Research Fund for the National Natural Science Foundation of China(Nos.61275067 and 61302026)the Provincial Natural Foundation of Jiangsu(Nos.BK2012830 and BK2012432)
文摘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.
基金supported by National Natural Science Foundation of China(10961021,11001222)
文摘In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd（M） = IFP-id（M＋） and IFP-fd（M＋）=IFP-id（M） for any R-module M over any ring R. Let ：Z-In （resp., ＂Zgv,~） he the class of all left （resp., right） R-modules of IFP-injective （resp., IFP-flat） dimension at most n. We prove that every right R-module has an IFn- preenvelope, （IFn,IF⊥n） is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id（RR） 〈 n, （IIn,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.
文摘In this paper, a flatly broadened and highly coherent supercontinuum generation by induced modulation instability in ANDi-PCF is proposed. The numerical results show that the bandwidth of supercontinuun can be increased by increasing the modulation depth and the coherence property of supercontinuum can be improved with the optimal modulation frequency. A highly coherent supercontinuum with the 10 dB bandwidth of 1305 nm and the flatness of less than 10 dB can be obtained when seeding at the modulation depth of 0.3 and modulation frequency of 24 THz.
基金Supported by NSFC(Grant No.11601304)Foundation for University Key Teacher by Henan Province(2019GGJS204)。
文摘LetRbeanarbitrary ring and let S be a separable and quasi-Frobenius extension of R.Then for any right S-module M,the Gorenstein flat dimensions of MS andMR areidentical.As a consequence,the Gorenstein weak global dimension of S is less than or equal to that of R.
基金supported by NSF of Guangxi Province of China(Grant No.2020GXNSFAA159120).
文摘Inspired by Bouchiba’s work on generalized Gorenstein flat modules,we introduce the generalized(F,A)-Gorenstein flat modules in this paper.By defining generalizedA-complete flat resolution,a new measure of the dimension of generalized(F,A)-Gorenstein flat modules is given.When(⊥1,)AAis a complete cotorsion pair,R is a right coherent ring andAcontains injective right R-modules,it is proved that for each M with finite flat dimension,the generalized(F,A)-Gorenstein flat dimension of M is equal to the(F,A)-Gorenstein flat dimension of M.
