关于半clean群环

环$R$称为是半clean的, 是指环中的每个元素都是一个单位与一个周期元的和. clean环是半clean的. 刻画半clean群环的一般情形是不容易的. 我们的目的是考虑如下问题：若$G$ 是局部有限群或者是阶是3的循环群, 群环$RG$何时是semiclean的. clean群环上的一些已有结果被推广.     

Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with $p^4$ elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.     

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A₂ (resp. A₁⁽¹⁾. We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.     

Multi‐shelled Hollow Metal–Organic Frameworks

Hollow metal–organic frameworks (MOFs) are promising materials with sophisticated structures, such as multiple shells, that cannot only enhance the properties of MOFs but also endow them with new functions. Herein, we show a rational strategy to fabricate multi‐shelled hollow chromium (III) terephthalate MOFs (MIL‐101) with single‐crystalline shells through step‐by‐step crystal growth and subsequent etching processes. This strategy relies on the creation of inhomogeneous MOF crystals in which the outer layer is chemically more robust than the inner layer and can be selectively etched by acetic acid. The regulation of MOF nucleation and crystallization allows the tailoring of the cavity size and shell thickness of each layer. The resultant multi‐shelled hollow MIL‐101 crystals show significantly enhanced catalytic activity during styrene oxidation. The insight gained from this systematic study will aid in the rational design and synthesis of other multi‐shelled hollow structures and the further expansion of their applications.     

Unconventional Route to Uniform Hollow Semiconducting Nanoparticles with Tailorable Dimensions,Compositions, Surface Chemistry,and Near‐Infrared Absorption

Despite impressive recent advances in the synthesis of lead chalcogenide solid nanoparticles, there are no examples of lead chalcogenide hollow nanoparticles (HNPs) with controlled diameter and shell thickness as current synthetic approaches for HNPs have inherent limitations associated with their complexity, inability to precisely control the dimensions, and limited possibilities with regard to applicable materials. Herein, we report on an unconventional strategy for crafting uniform lead chalcogenide (PbS and PbTe) HNPs with tailorable size, surface chemistry, and near‐IR absorption. Amphiphilic star‐like triblock copolymers [polystyrene‐block‐poly(acrylic acid)‐block‐polystyrene and polystyrene‐block‐poly(acrylic acid)‐block‐poly(3,4‐ethylenedioxythiophene)] were rationally synthesized and exploited as nanoreactors for the formation of uniform PbS and PbTe HNPs. Compared to their solid counterparts, the near‐IR absorption of the HNPs is blue‐shifted owing to the hollow interior. This strategy can be readily extended to other types of intriguing low‐band‐gap HNPs for diverse applications.     

Development of a novel hollow‐fiber liquid‐phase microextraction based on oil‐in‐salt and its comparison with conventional one

A novel hollow‐fiber liquid‐phase microextraction based on oil‐in‐salt was proposed and introduced for the simultaneous extraction and enrichment of the main active compounds of hesperidin, honokiol, shikonin, magnolol, emodin, and β,β′‐dimethylacrylshikonin in a formula of Zi‐Cao‐Cheng‐Qi decoction and the single herb, Fructus Aurantii Immaturus , Cortex Magnoliae Officinalis , Radix et Rhizoma , and Lithospermum erythrorhizon , composing the formula prior to their analysis by high‐performance liquid chromatography. The results obtained by the proposed procedure were compared with those obtained by conventional hollow‐fiber liquid‐phase microextraction, and the proposed procedure mechanism was described. In the procedure, a hollow‐fiber segment was first immersed in organic solvent to fill the solvent in the fiber lumen and wall pore, and then the fiber was again immersed into sodium chloride solution to cover a thin salt membrane on the fiber wall pore filling organic solvent. Under the optimum conditions, the enrichment factors of the analytes were 0.6–109.4, linearities were 0.002–12 μg/mL with r ² ≥ 0.9950, detection limits were 0.6–12 ng/mL, respectively. The results showed that oil‐in‐salt hollow‐fiber liquid‐phase microextraction is a simple and effective sample pretreatment procedure and suitable for the simultaneous extraction and concentration of trace‐level active compounds in traditional Chinese medicine.     

Enhanced Fluorescence Imaging and Photodynamic Cancer Therapy Using Hollow Mesoporous Nanocontainers

Here we show that “off‐on” type of photodynamic therapy agents could be developed using hollow mesoporous silica nanoparticles (HMSNPs), which can be used not only for enhancing delivery of photosensitizers to cancer cells but also for enabling switchable optical properties of the photosensitizers. Fluorescence and singlet oxygen generation of the photosensitizer‐loaded HMSNP are turned off in its native state. In vitro cell studies showed that this HMSNP‐based “off‐on” agent may have potential utility in selective fluorescence detection and photodynamic therapy of cancers.     

Controlled Synthesis of Hollow PbS‐TiO2 Hybrid Structures through an Ion Adsorption–Heating Process and their Photocatalytic Activity

Hollow hybrid nanostructures have received significant attention because of their unique structural features. This study reports a facile ion adsorption–heating method to fabricate hollow PbS‐TiO₂ hybrid particles. In this method, the TiO₂ spheres used as a substrate material to grow PbS are aggregates of many small amorphous TiO₂ particles, and each small particle is covered with thioglycolic acid ligands through Ti⁴⁺–carboxyl coordination. When Pb²⁺ ions are added to a colloidal solution of these TiO₂ spheres, these ions are adsorbed by sulfhydryl (‐SH) groups to form metal thiolates, and the C−S bond is dissociated by heating to release S²⁻. The S²⁻ ions react with Pb²⁺ ions to form PbS without additive sulfur sources. Additionally, the amorphous TiO₂ spheres are transformed into the anatase phase during the heating process. As a result, the crystallization of TiO₂ spheres along with the formation of PbS is simultaneously carried out by heating. During the heating process, owing to the Kirkendall effect of S²⁻ diffusion and the Ostwald ripening effect of the crystallization of amorphous TiO₂ spheres, PbS‐TiO₂ hollow hybrid structures can be obtained. The XRD and XPS characterizations proved the formation of anatase TiO₂ and PbS. The TEM characterization confirmed the formation of hollow structures in the PbS‐TiO₂ hybrid sample. The photocatalytic activity of the hollow PbS‐TiO₂ hybrid spheres have been investigated for the degradation of Cr⁶⁺ under visible light. The results show that hollow PbS‐TiO₂ hybrid spheres exhibited the highest photocatalytic activity, in which almost all the Cr⁶⁺ was degraded after 140 min.     

On the inheritance of the strongly π-extending property

In this article, we focus on modules with the property that every projection invariant submodule is essential in a fully invariant direct summand. In contrast to π-extending condition, it is shown that the former property is inherited by direct summands and Morita invariant. An application of our results yields that the endomorphism ring of a free module enjoys the property. Moreover, we characterize generalized triangular matrix rings with the aforementioned property and apply to somewhat special cases.     

Wedge Operations and Doubling Operations of Real Toric Manifolds

This paper deals with two things.First,the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G.Second,the author focuses on a specific canonical extensions called doublings and presents their various properties.They include existence of infinitely many real topological toric manifolds admitting complex structures,and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups.Moreover,some questions about real topological toric manifolds related to Halperin's toral rank conjecture are presented.