The Report of Homological Algebra

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.

Injection modulation of p^+-n emitter junction in 4H-SiC light triggered thyristor by double-deck thin n-base

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.

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.

EXPLICIT DESCRIPTION OF A CLASS OF INDECOMPOSABLE INJECTIVE MODULES

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.

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.

On Pseudo Small Injective Modules

The definition of principally pseudo injectivity motivates us to generalize tae notion of injectivity, noted SP pseudo injectivity. The aim of this paper is to investigate characterizations and properties of SP pseudo injective modules. Various results are devel- oped, many extending known results. As applications, we give some characterizations on Noetherian rings, QI rings, quasi-Frobenius rings.

Simplified modeling of frequency behavior in photonic crystal vertical cavity surface emitting laser with tunnel injection quantum dot in active region

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.

Relative(n,k)-Weak Cotorsion Modules

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.

Generalized Regular Ring

Using module class C R=Mx∈M,xRT=0,T∈I , we introduced the concepts of C R finitely generated module, C R finitely presented module and C R regular ring. We also discussed the criterion for C R regular ring,and the relations between C R regular ring and C R FP injective module.

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.

C3-Modules

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.

ω-Gorenstein injective modules and dimensions

Gorenstein injective modules and dimensions have been studied extensively by many authors. In this paper, we investigate Gorenstein injective modules and dimensions relative to a Wakamatsu tilting module.

Generalized Gorenstein Modules

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.

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

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. ≤

Relative T-Injective Modules and Relative T-Flat Modules

Abstract Let T be a Wakamatsu tilting module. A module M is called （n, T）-copure injective （resp. （n, T）-copure flat） if εT^1（N, M） = 0 （resp. Г1^T（N, M） = 0） for any module N with T-injective dimension at most n （see Definition 2.2）. In this paper, it is shown that M is （n, T）-copure injective if and only if M is the kernel of an In（T）-precover f ： A → B with A ∈ ProdT. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is （n, T）-copure injective.

V-Gorenstein Injective Modules Preenvelopes and Related Dimension

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.

FPn-Injective, FPn-Flat Covers and Preenvelopes, and Gorenstein AC-Flat Covers

We prove that, for any n 〉 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules （i.e., absolutely clean and, respectively, level modules）. Then we consider a generalization of the class of （strongly） Gorenstein flat modules, i.e., the （strongly） Gorenstein AC-flat modules （cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module）. We prove that some of the properties of Gorenstein fiat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that （as in the case of Gorenstein flat modules） every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.

A Note on DG-Gorenstein Injective Complexes

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.

On Ding Projective Complexes

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.