On the Schröder–Bernstein property for modules

The well known Schröder–Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. Kaplansky asked whether direct summands subisomorphic abelian groups are always isomorphic? The question generated a great deal of interest. The study of this question for the general class of modules has been somewhat limited. We extend the study of this question for modules in this paper. We say that a module Msatisfies the Schröder–Bernstein property (S-B property) if any two direct summands of M which are subisomorphic to direct summands of each other, are isomorphic. We show that a large number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. It is also shown that over a Noetherian ring R, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings R for which every R-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain R, any two quasi-continuous subisomorphic R-modules are isomorphic if and only if R is a PID. We study other conditions related to the S-B property and obtain characterizations of certain classes of rings via those conditions. Examples which delimit and illustrate our results are provided.     

Studies on photoinduced effects in pulse-electrodeposited Ag/Hg-1212/CdSe hetero-nanostructures

Metal/superconductor/semiconductor (Ag/Hg-1212/CdSe) hetero-nanostructures have been fabricated using pulse-electrodeposition technique and are characterized by X-ray diffraction (XRD), full-width at half-maximum (FWHM) and scanning electron microscopy (SEM) studies. The junction capacitance of Ag/Hg-1212, Hg-1212/CdSe and Ag/Hg-1212/CdSe heterojunctions is measured in dark and under laser irradiation at room temperature. The nature of the junction formed and built-in-junction potentials were determined. The increase in carrier concentration across the junction due to photo-irradiation has been observed.     

Effect of laser irradiation of C-V characteristics of electrodeposited Ag/Tl-2223/CdSe hetero-nanostructures

One of the innovative technological directions for the high-temperature superconductors has been persued by fabricating the heteroepitaxial multilayer structures such as superconductor-semiconductor heterostructures. In the present investigation, metal/superconductor/semiconductor (Ag/Tl-2223/CdSe) hetero-nanostructures have successfully been fabricated using dc electrodeposition technique and were characterized by X-ray diffraction (XRD), full-width at half-maximum (FWHM) and scanning electron microscopy (SEM) studies. The measurement of junction capacitance as a function of biasing voltage was used for the estimation of junction built-in-potential (V _D) and to study the charge distribution in a heterojunction. The Mott-Schottky plots were measured for each junction in dark and under the photo-irradiation. The effect of laser irradiation on C-V characteristics of hetero-nanostructure has been studied.     

AnomalousK-conversion coefficient of the 124keV transition in131Cs

The absoluteK-conversion coefficients of the 496 keV-124 keV cascadeγ-rays in¹³¹Cs have been determined from angular correlation experiments using a new approach. The value ofα _K =0.0100±0.0007 obtained for the 496 keV transition is in excellent agreement with the theory whereas a strong case is presented for the existence of an anomalous nuclear-structure-dependent conversion coefficient for the 124 keV transition. TheK-conversion coefficient for this transition has been measured to be 0.742±0.077 giving a value of 1.25±0.13 for the ratioα _K ^(expet) /α _K ^(theory) . The result is interpreted as being due to the presence of electric monopole,E0, electron transition. A value of 0.26±0.14 has been obtained for theE0/E2 electron conversion ratio and the absolute value of the electric monopole matrix element has been determined to be 0.07±0.02.     

在不同温度下球墨鑄铁的冲击性能

本文描述高温冲击实验的技术,以及由-85℃至1070℃的铸态和锻造后的球墨铸铁的冲击性能。研究结果找到了基体主要为铁素体的球墨铸铁的两个变脆温度范围(-20℃以下和550℃至700℃之间)和一个比较具有韧性的温度范围(850℃与1050℃之间)。在这些温度范围内,曾考查试样的断口情况和断裂时的弯曲角度,并与其基体组织的变化相联系,进行讨论。最后,本文指出研究结果在球墨铸铁的应用上所起的作用,以及在球墨铸铁热加工时对于选择热加工温度的意义。     

Stimulated scattering of light in a laser produced magnetoplasma

This note considers stimulated scattering of laser light in the ordinary mode by magnetosonic waves in a plasma. Problems of three-wave decay interaction and modulation instability are investigated. The growth rates and thresholds are obtained analytically. Application to laser-plasma interaction is discussed.     

Replacement of native copper in superoxide dismutase with copper-64 without much loss in its enzymatic activity

Superoxide dismutase, containing copper and zinc, has been labeled with copper-64 by incubating the prepared apoenzyme with cupric(65) chloride at room temperature. No significant loss in the enzymatic activity was observed after labeling. The incorporation of copper-64 was ascertained by starch gel electrophoresis and high performance liquid chromatography. The labeling efficiency was found to be >95%.     

Quasi-Baer module hulls and applications

Let V be a module with $S=\mathrm{End}(V)$. V is called a quasi-Baer module if for each ideal J of S, $r_V(J)=eV$ for some $e^2=e\in S$. On the other hand, V is called a Rickart module if for each $?\in S$, $\mathrm{Ker}(?)=eV$ for some $e^2=e\in S$. For a module N, the quasi-Baer module hull $\mathbf{qB}(N)$ (resp., the Rickart module hull $\mathbf{Ric}(N)$) of N, if it exists, is the smallest quasi-Baer (resp., Rickart) overmodule, in a fixed injective hull $E(N)$ of N. In this paper, we initiate the study of quasi-Baer and Rickart module hulls. When a ring R is semiprime and ideal intrinsic over its center, it is shown that every finitely generated projective R-module has a quasi-Baer hull. Let R be a Dedekind domain with F its field of fractions and let $\{K_i|i\in \mathrm{\Lambda }\}$ be any set of R-submodules of $F_R$. For an R-module $M_R$ with ${\mathrm{Ann}}_R(M)\ne 0$, we show that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has a quasi-Baer module hull if and only if $M_R$ is semisimple. This quasi-Baer hull is explicitly described. An example such that $M_R\oplus {({?}_{i\in \mathrm{\Lambda }}{K}_i)}_R$ has no Rickart module hull is constructed. If N is a module over a Dedekind domain for which $N/t(N)$ is projective and ${\mathrm{Ann}}_R(t(N))\ne 0$, where $t(N)$ is the torsion submodule of N, we show that the quasi-Baer hull $\mathbf{qB}(N)$ of N exists if and only if $t(N)$ is semisimple. We prove that the Rickart module hull also exists for such modules N. Furthermore, we provide explicit constructions of $\mathbf{qB}(N)$ and $\mathbf{Ric}(N)$ and show that in this situation these two hulls coincide. Among applications, it is shown that if N is a finitely generated module over a Dedekind domain, then N is quasi-Baer if and only if N is Rickart if and only if N is Baer if and only if N is semisimple or torsion-free. For a direct sum $N_R$ of finitely generated modules, where R is a Dedekind domain, we show that N is quasi-Baer if and only if N is Rickart if and only if N is semisimple or torsion-free. Examples exhibiting differences between the notions of a Baer hull, a quasi-Baer hull, and a Rickart hull of a module are presented. Various explicit examples illustrating our results are constructed.     

Solvability of a class of boundary value problems in the space of convergent sequences

We consider two types of infinite systems of second-order differential equations with boundary conditions in the Banach sequence space c. For each system, sufficient conditions are provided for the existence of solutions. Our approach is based on the use of measure of noncompactness concept and the application of Darbo’s theorem for condensing operators. Some examples are presented to illustrate the obtained results.