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
Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module...Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.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 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 t...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.展开更多
To overcome hole-injection limitation of p^+-n emitter junction in 4H-SiC light triggered thyristor, a novel high- voltage 4H-SiC light triggered thyristor with double-deck thin n-base structure is proposed and demon...To overcome hole-injection limitation of p^+-n emitter junction in 4H-SiC light triggered thyristor, a novel high- voltage 4H-SiC light triggered thyristor with double-deck thin n-base structure is proposed and demonstrated by two- dimensional numerical simulations. In this new structure, the conventional thin n-base is split to double-deck. The hole- injection of p^+-n emitter junction is modulated by modulating the doping concentration and thickness of upper-deck thin n- base. With double-deck thin n-base, the current gain coefficient of the top pnp transistor in 4H-SiC light triggered thyristor is enhanced. As a result, the triggering light intensity and the turn-on delay time of 4H-SiC light triggered thyristor are both reduced. The simulation results show that the proposed 10-kV 4H-SiC light triggered thyristor is able to be triggered on by 500-mW/cm^2 ultraviolet light pulse. Meanwhile, the turn-on delay time of the proposed thyristor is reduced to 337 ns.展开更多
This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows th...This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤展开更多
Let R and S be associative rings and sVR a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR(Zv(R),-) and HomR(-,Zv(R)) exact exact complex . of V-inject...Let R and S be associative rings and sVR a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR(Zv(R),-) and HomR(-,Zv(R)) exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v(R) which leads to the fact that V-Gorenstein injective modules admit exact right Iv (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv (R) (I, N) = 0 for all modules I with finite Iv (R)-injective dimension.展开更多
In this work, the characteristics of the photonic crystal tunneling injection quantum dot vertical cavity surface emitting lasers(Ph C-TIQD-VCSEL) are studied through analyzing a modified modulation transfer functio...In this work, the characteristics of the photonic crystal tunneling injection quantum dot vertical cavity surface emitting lasers(Ph C-TIQD-VCSEL) are studied through analyzing a modified modulation transfer function. The function is based on the rate equations describing the carrier dynamics at different energy levels of dot and injector well. Although the frequency modulation response component associated with carrier dynamics in wetting layer(WL) and at excited state(ES) levels of dots limits the total bandwidth in conventional QD-VCSEL, our study shows that it can be compensated for by electron tunneling from the injector well into the dot in TIQD structure. Carrier back tunneling time is one of the most important parameters, and by increment of that, the bias current dependence of the total bandwidth will be insignificant. It is proved that at high bias current, the limitation of the WL-ES level plays an important role in reducing the total bandwidth and results in rollovers on 3-d B bandwidth-I curves. In such a way, for smaller air hole diameter of photonic crystal, the effect of this reduction is stronger.展开更多
The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand ...The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand any morphism f:E→G is null homotopic whenever E is a DG-injective complex.展开更多
One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summa...One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.展开更多
We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply...We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.展开更多
Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group P, called the injective complete cohomological dimension of F, denoted by iccd P. We study this dimension and ...Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group P, called the injective complete cohomological dimension of F, denoted by iccd P. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group F, which is denoted by Gid F. We show that Gid F is related to iccd F, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd P is a refinement of Gid P. In addition, we show that silp F = spli F 〈 ∞if and only if the Shapiro lemma holds for injective complete cohomology.展开更多
The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modu...The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modules.展开更多
Let R be a ring and n,k be two non-negative integers.As an extension of several known notions,we introduce and study(n,k)-weak cotorsion modules using the class of right R-modules with n-weak fat dimensions at most k....Let R be a ring and n,k be two non-negative integers.As an extension of several known notions,we introduce and study(n,k)-weak cotorsion modules using the class of right R-modules with n-weak fat dimensions at most k.Various examples and applications are also given.展开更多
In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-mod...In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ Z.展开更多
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...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).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This research is in part supported by a grant from IPM.
文摘Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.
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This work was partially supported by NSFC(Grant Nos.11571164 and 11571341).
文摘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.
基金supported by the National Natural Science Foundation of China(Grant No.51677149)
文摘To overcome hole-injection limitation of p^+-n emitter junction in 4H-SiC light triggered thyristor, a novel high- voltage 4H-SiC light triggered thyristor with double-deck thin n-base structure is proposed and demonstrated by two- dimensional numerical simulations. In this new structure, the conventional thin n-base is split to double-deck. The hole- injection of p^+-n emitter junction is modulated by modulating the doping concentration and thickness of upper-deck thin n- base. With double-deck thin n-base, the current gain coefficient of the top pnp transistor in 4H-SiC light triggered thyristor is enhanced. As a result, the triggering light intensity and the turn-on delay time of 4H-SiC light triggered thyristor are both reduced. The simulation results show that the proposed 10-kV 4H-SiC light triggered thyristor is able to be triggered on by 500-mW/cm^2 ultraviolet light pulse. Meanwhile, the turn-on delay time of the proposed thyristor is reduced to 337 ns.
基金the National Natural Science Foundation of China (No.10171082) the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China and NWNU-KJCXGC212.
文摘This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤
文摘Let R and S be associative rings and sVR a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a HomR(Zv(R),-) and HomR(-,Zv(R)) exact exact complex . of V-injective modules Ii and Ii,i ∈ N0, such that N We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class ,4v(R) which leads to the fact that V-Gorenstein injective modules admit exact right Iv (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V- Gorenstein injective if and only if N @ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Corenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext Iv (R) (I, N) = 0 for all modules I with finite Iv (R)-injective dimension.
文摘In this work, the characteristics of the photonic crystal tunneling injection quantum dot vertical cavity surface emitting lasers(Ph C-TIQD-VCSEL) are studied through analyzing a modified modulation transfer function. The function is based on the rate equations describing the carrier dynamics at different energy levels of dot and injector well. Although the frequency modulation response component associated with carrier dynamics in wetting layer(WL) and at excited state(ES) levels of dots limits the total bandwidth in conventional QD-VCSEL, our study shows that it can be compensated for by electron tunneling from the injector well into the dot in TIQD structure. Carrier back tunneling time is one of the most important parameters, and by increment of that, the bias current dependence of the total bandwidth will be insignificant. It is proved that at high bias current, the limitation of the WL-ES level plays an important role in reducing the total bandwidth and results in rollovers on 3-d B bandwidth-I curves. In such a way, for smaller air hole diameter of photonic crystal, the effect of this reduction is stronger.
基金This work was supported by the National Natural Science Foundation of China(grant no.11501451)the Funds for Talent Introduction of Northwest Minzu University(grant no.XBMUYJRC201406).
文摘The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand any morphism f:E→G is null homotopic whenever E is a DG-injective complex.
文摘One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A A B = 0, then A B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.
文摘We introduce a generalization of the Gorenstein injective modules:the Gorenstein FPn-injective modules(denoted by GI_(n)).They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-),with A any FP_(n)-injective module.Thus,GL_(o)is the class of classical Gorenstein injective modules,and GI_(1)is the class of Ding injective modules.We prove that over any ring R,for any n≥2,the class GI_(n)is the right half of a perfect cotorsion pair,and therefore it is an enveloping class.For n=1 we show that GI_(1)(i.e.,the Ding injectives)forms the right half of a hereditary cotorsion pair.If moreover the ring R is coherent,then the Ding injective modules form an enveloping class.We also define the dual notion,that of Gorenstein FP_(n)-projectives(denoted by GP_(n)).They generalize the Ding projective modules,and so,the Gorenstein projective modules.We prove that for any n≥2 the class GP_(n)is the left half of a complete hereditary cotorsion pair,and therefore it is special precovering.
文摘Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group P, called the injective complete cohomological dimension of F, denoted by iccd P. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group F, which is denoted by Gid F. We show that Gid F is related to iccd F, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd P is a refinement of Gid P. In addition, we show that silp F = spli F 〈 ∞if and only if the Shapiro lemma holds for injective complete cohomology.
文摘The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modules.
文摘Let R be a ring and n,k be two non-negative integers.As an extension of several known notions,we introduce and study(n,k)-weak cotorsion modules using the class of right R-modules with n-weak fat dimensions at most k.Various examples and applications are also given.
基金Supported by National Natural Science Foundation of China(Grant Nos.11561039 and 11761045)Natural Science Foundation of Gansu Province of China(Grant No.17JR5RA091)
文摘In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each Xm is a Ding projective R-module for all m ∈ Z.
基金S.Estrada was partly supported by grant MTM2016-77445-PFEDER funds and by grant 19880/GERM/15 from the Fundacion Seneca-Agencia de Ciencia y Tecnologfa de la Region de Murcia.
文摘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).