Injective Modules Research Articles

Mid-infrared high repetition mode-locked laser based on cross-band all-optical injection modulation

High speed pulse generation with adjustable output characteristics in the mid-infrared (MIR) wavelength is a challenge due to the lack of high speed optoelectronic modulator operating at this band. In this paper, we proposed a MIR actively mode-locked laser based on cross-band all-optical injection modulation. The proposed laser is modulated by injecting 1.55-μm pump pulses into the specially-designed cross-band group-velocity-matched fiber. By numerical simulation, we studied in detail the evolution mechanism of the pulses and the influence of the cavity parameters on the output pulse features. It is shown that the laser is suitable for high repetition rate pulse generation up to 40 GHz at 2.771 μm, and the pulse shape can be flexibly adjusted. The proposed MIR laser has great potential in MIR free-space optical (FSO) communication.

Completeness and stability of the class of injective $S$-acts

We deal with questions concerning the completeness and stability of a class of injective acts and a class of weakly injective acts over a monoid S. The concepts of an injective S-act and of a weakly injective S-act are analogs of the concept of an injective module. In the theory of modules, the corresponding notions of injectivities in accordance with Baer’s criterion coincide. Also we will look into the completeness and stability of a class of principally weakly injective S-acts and a class of fg-weakly injective S-acts, which are analogs of p-injective modules and finitely injective modules.

On the Weak-Injectivity Profile of a Ring

We consider two types of weak-injectivity domains of a module. Small weak-injectivity domains gauge the extent of weak injectivity of modules; weak-injectivity domains are an alternative way to gauge their injectivity. We focus on notions which are opposite, respectively, to weakly injective or injective modules according to these two schemes. While we do consider both notions to some extent, our emphasis is on modules with smallest weak-injectivity domain; we name those modules extremely poor. The intersection of all (small) domains of weak injectivity is the natural focus of attention. The intersection of all (small) domains of weak injectivity contains the class of all (finitely generated) semisimple modules, but it is not always equal to it. The domain of weak injectivity of an extremely poor module thusly depends on the ring in question. We consider necessary and sufficient conditions for said intersections to consist solely of semisimple modules and look at examples of both circumstances, when this is the case and when it is not. When the intersection of weak-injectivity domains consists of semisimple modules, every extremely poor module is poor; we study further connections between these two notions. Comparisons are also drawn to paupers (poor modules with no proper poor direct summands); paupers are a type of poor module that is significant in understanding the intrinsic structure of poor modules. Extremely poor modules over PCI domains and extremely poor abelian groups are given particular attention.

On secondary and representable modules over almost Dedekind domains

Matlis showed that an injective module over a commutative Noetherian ring R can be completely decomposed as a direct sum of indecomposable injective submodules. In this paper, we prove the Matlis’ Theorem for almost Dedekind domains. Then we characterize the secondary modules and classify the indecomposable secondary modules over almost Dedekind domains. Also we prove every P-secondary module over an almost Dedekind domain is pure-injective, where Finally, we characterize the representable finitely generated modules over almost Dedekind domains.

Optimal Neutral-Point Voltage Balancing Algorithm for Three-Phase Three-Level Converters With Hybrid Zero-Sequence Signal Injection and Virtual Zero-Level Modulation

This article presents an optimal carrier-based voltage balancing scheme for three-phase three-level converters. The proposed scheme utilizes two available degrees of freedom, i.e., zero-sequence signal injection and virtual zero-level modulation (VZM), to eliminate the low-frequency neutral-point voltage oscillation. It is universally effective over the full power factor and modulation index range and easy to implement in digital controllers. The hybrid algorithm combines the merits of both approaches, which offers the optimal performance regarding controllability, switching device power losses, and output harmonics. The main drawbacks of VZM, i.e., the increased switching loss and high-frequency harmonics due to additional switching transitions, have been minimized in the proposed scheme. The performance of the proposed scheme is evaluated through simulation and experiment.

Torsion of injective modules and weakly pro-regular sequences

Let R a commutative ring, an ideal, I an injective R-module and a multiplicatively closed set. When R is Noetherian it is well-known that the -torsion sub-module the factor module and the localization IS are again injective R-modules. We investigate these properties in the case of a commutative ring R by means of a notion of relatively--injective R-modules. In particular we get another characterization of weakly pro-regular sequences in terms of relatively injective modules. Also we present examples of non-Noetherian commutative rings R and injective R-modules for which the previous properties do not hold. Moreover, under some weak pro-regularity conditions we obtain results of Mayer-Vietoris type.

On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings

Let $R$ be a ring and $\mathscr{C}$ be a class of some finitely presented left $R$-modules. A left $R$-module $M$ is called $\mathscr{C}$-injective if Ext$^1_R(C, M)=0$ for every $C\in \mathscr{C}$; a left $R$-module $M$ is called $\mathscr{C}$-projective if ${\rm Ext}^1_R(M, E)=0$ for any $\mathscr{C}$-injective module $E$. $R$ is called left $\mathscr{C}$-coherent if every $C\in \mathscr{C}$ is 2-presented; $R$ is called left strongly $\mathscr{C}$-coherent, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ and $P$ is finitely generated projective, then $K$ is $\mathscr{C}$-projective; a ring $R$ is called left $\mathscr{C}$-semihereditary, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ , $P$ is finitely generated projective, then $K$ is projective. In this paper, we give some new characterizations and properties of left $\mathscr{C}$-coherent rings, left strongly $\mathscr{C}$-coherent rings and left $\mathscr{C}$-semihereditary rings.

Singular compactness and definability for $$\Sigma $$-cotorsion and Gorenstein modules

We introduce a general version of singular compactness theorem which makes it possible to show that being a $\Sigma$-cotorsion module is a property of the complete theory of the module. As an application of the powerful tools developed along the way, we give a new description of Gorenstein flat modules which implies that, regardless of the ring, the class of all Gorenstein flat modules forms the left-hand class of a perfect cotorsion pair. We also prove the dual result for Gorenstein injective modules.

Torque Ripple Suppression of Permanent Magnet Synchronous Motor Based On Robust Current Injection

In order to improve the performance of permanent magnet synchronous motor (PMSM) servo system and suppress the influence of torque ripple on the control system, a robust current injection control method combining active disturbance rejection controller (ADRC) and current injection is proposed. We designed a current injection module to suppress the torque ripple of the motor and proposed an ADRC to improve the anti-disturbance performance of the system, which ensures the strong robustness and rapidity of the system. The simulation results show that compared with the conventional PI control and PI superimposed current injection, the proposed method can more effectively suppress the torque ripple, while improving the robustness and dynamic response of the system.

Some characterizations of coherent rings in terms of strongly FP-injective modules

An R-module M is called strongly FP-injective if for any finitely presented R-module P and any i > 0. Denoted by the class of all strongly FP-injective R-modules and by the left orthogonal class with respect to Comparing with some classical results of Noetherian rings, we show that a ring is coherent if and only if is closed under pure submodules; if and only if any finitely generated module in is finitely presented; if and only if R is -coherent and is closed under direct sums; if and only if for any nonnegative integer n, any finitely presented left R-module X, any R-T-bimodule M and any injective right T-module E. In addition, we show that a module is injective if and only if it is a pure -periodic module, where denotes the class of all injective modules.

Automorphism-Extendable and Endomorphism-Extendable Modules

This review paper is concerned with modules in which all automorphisms (endomorphisms) of submodules can be extended to endomorphisms of the entire module. We consider old results and obtain a number of new results. We also consider automorphism-invariant, quasi-injective, and injective modules.

On $$\varvec{\mathcal {D}}$$-closed submodules

A submodule N of a module M is called $$\mathcal {D}$$-closed if the socle of M / N is zero. $$\mathcal {D}$$-closed submodules are similar to $$\mathcal {S}$$-closed submodules (a generalization of closed submodules) defined through nonsingular modules. First, we describe the smallest proper class (due to Buchsbaum) containing the class of short exact sequences determined by $$\mathcal {D}$$-closed submodules in terms of that submodule, and show that it coincides with other classes of modules under certain conditions. Second, we study coprojective modules of this class, called edc-flat modules. We give some equivalent conditions for injective modules to be edc-flat for special rings, and for edc-flat modules to be projective (flat) for any ring.

Strongly injective modules

Let R be a ring with identity and M be a left R-module. The module M is called strongly injective if whenever $$M+K=N$$ with $$M\subseteq N$$ , there exists a submodule $$K^{'}$$ of K such that $$M\oplus K^{'}=N$$ . In this paper, we provide the various properties of the class of these modules. In particular, we prove that M is strongly injective if and only if it is semisimple injective. Moreover, we give new characterizations of semisimple rings and left V-rings via strongly injective modules. Finally, we show that every strongly injective module is strongly noncosingular.

On a generalization of self-injective rings

In this work the notion of left (right) self-injective ring is generalized. We consider rings that are direct sum of injective module and semisimple module as a left (respectively, right) module above itself. We call such rings left (right) semi-injective and research their properties with the help of two-sided Peirce decomposition of the ring. The paper contains the description of left noetherian left semi-injective rings. It is proved that any such ring is a direct product of (two-sided) self-injective ring and several quotient rings (of special kind) of rings of upper triangular matrices over skew fields. From this description it follows that for left semi-injective rings we have the analogue of the classical result for self-injective rings. Namely, if a ring is left noetherian and left semi-injective then this ring is also right semi-injective and two-sided artinian.

TAIZAC -TAndem Injectors Zapping ACtivation- for Thermal Efficiency Improvement of Diesel Engine

<div class="section abstract"><div class="htmlview paragraph">In order to explore the potential of thermal efficiency improvement of diesel engines with injection rate shaping, TAIZAC (TAndem Injectors Zapping ACtivation) injector consisting of commercially available two common-rail injectors was developed in-house. A simple architecture adopted for TAIZAC injector consists of an upper injector connected to a high pressure source, a lower injector injecting into the cylinder and sealing parts between the two injectors. A fast response piezo-actuated injector (G3P) was employed as the upper one with an additionally drilled large axial orifice for faster fuel supply. A solenoid-actuated injector (G4S) was employed as the lower one for its applicability to inner volume reduction by body cut. Control of injection timing and duration of the upper injector realized not only the originally targeted inversed-delta injection rate shaping, but also sharper rise of initial injection rate and modulation of injection pressure among and during multi-injection pulses. Single cylinder diesel engine performance tests were conducted to demonstrate the effects of a combination of 100MPa low pressure pre and pilot injections, sharper rise of initial injection rate and inversed-delta rate shaping for 200MPa high pressure main injection. Compared to the case of conventional G4S injector, TAIZAC injector case exhibited 1.4pt improvement of indicated thermal efficiency mainly due to 2.3pt reduced cooling loss.</div></div>

A New Generalization of Injective Modules

In this paper, as a generalization of injective modules, we define two different modules: modules that have the property (δ-SE) and modules that have the property (δ-SSE), and we investigate basic properties of them. Namely, modules that have a δ-supplement that is a direct summand in its every extension and modules that have a strong δ-supplement in its every extension are tackled here. Particularly, it is proved that a ring whose modules have the property (δ-SE) is δ-semiperfect. Let R be a ring, M be an R-module with IM=0 for an ideal of R. It is shown that if the R-module M has the property (δ-SE), then so does ¯R-module M, under a special condition, where ¯R=R/I. Finally we supply an example showing that a module that has the property (δ-SE) may not have the property (δ-SSE).

ON RICKART MODULES

Gangyong Lee, S.Tariq Rizvi, and Cosmin S.Roman studied Rickart modules.
 The main purpose of this paper is to develop the properties of Rickart modules .
 We prove that each injective and prime module is a Rickart module. And we give characterizations of some kind of rings in term of Rickart modules.

Cotilting sheaves on Noetherian schemes

We develop theory of (possibly large) cotilting objects of injective dimension at most one in general Grothendieck categories. We show that such cotilting objects are always pure-injective and that they characterize the situation where the Grothendieck category is tilted using a torsion pair to another Grothendieck category. We prove that for Noetherian schemes with an ample family of line bundles a cotilting class is closed under injective envelopes if and only if it is invariant under twists by line bundles, and that such cotilting classes are parametrized by specialization closed subsets disjoint from the associated points of the scheme. Finally, we compute the cotilting sheaves of the latter type explicitly for curves as products of direct images of indecomposable injective modules or completed canonical modules at stalks.

Modules close to the automorphism-invariant and coinvariant

The aim of this paper is to introduce a general setting where some well-known results on essentially injective modules, automorphism-(co)invariant modules and small projective modules can be obtained by developing a general theory of modules which are (co)invariant under automorphisms of their covers and envelopes.

Experimental and numerical simulation of supercritical CO2 microbubble injection into a brine-saturated porous medium

Abstract Microbubbles are one solution to the channeling problem in the displacement process through porous media. However, applications of microbubbles to geological carbon sequestration (GCS) is scarcely reported. In this work, we injected supercritical CO2 (scCO2) into brine-saturated sandstone and mapped the displacement process by medical computed tomography (CT). The injections were performed by two methods: an analog of a regular drainage process and a special injection module incorporating microbubble injection. The microbubble injection increased the scCO2 saturation, which corresponds to the sweep efficiency, by alleviating the channeling effect. We then investigated the microbubble injection in numerical simulations and gained insights into the injection mechanism. Both injection processes were reasonably produced by assigning the sub-core permeability approximation with the porosity−capillary pressure approach ( ϕ - P c ). Based on these results, we attributed the macroscale effects of microbubble injection to the lowered entry capillary pressure.