Correspondences of Coclosed Submodules

We establish an order-preserving bijective correspondence between the sets of coclosed elements of some bounded lattices related by suitable Galois connections. As an application, we deduce that if M is a finitely generated quasi-projective left R-module with S = End _R (M) and N is an M-generated left R-module, then there exists an order-preserving bijective correspondence between the sets of coclosed left R-submodules of N and coclosed left S-submodules of Hom _R (M, N).     

Modules with Chain Conditions on Non-essential Submodules

Abstract

We investigate when modules which satisfy the descending (respectively, ascending) chain condition on non-essential submodules are uniform or Artinian (respectively, Noetherian).     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

Topology-like dynamical behavior of magnetization reversal in exchange-bias systems

In an exchange-bias system, the barriers with intrinsic potential energy may be asymmetric due to unidirectional anisotropy. Based on the Stoner-Wohlfarth model, we show that the asymmetric barriers may lead to four kinds of dynamical process underlying the hysteresis-loop measurement. These kinds of dynamical processes are different in a topology-like property, which can be controlled by the orientation of the external field. In our study, a new analysis approach has been proposed to reveal the dynamical behaviors of magnetization reversal. With this approach, coercivity, exchange-bias field, and asymmetry of hysteresis loops can be quantitatively obtained.     

Some Remarks on Generalized GCD Domains

An integral domain R is a generalized GCD (GGCD) domain if the semigroup of invertible ideals of R is closed under intersection. In this article we extend the definition of PF-prime ideals to GGCD domains and develop a theory of these ideals which allows us to characterize Prüfer and π -domains among GGCD domains. We also introduce the concept of generalized GCD modules as a natural generalization of GGCD domains to the module case. An R-module M is a GGCD module if the set of invertible submodules of M is closed under intersection. We show that an integral domain R is a GGCD domain if and only if a faithful multiplication R-module M is a GGCD module. Various properties and characterizations of faithful multiplication GGCD modules over integral domains are considered and consequently, necessary and sufficient conditions for a ring R(M), the idealization of M, to be a GGCD ring are given.     

Synthesis of intrinsically flame‐retardant copolyamides and their employment in PA6‐fibers

Flame‐retarded polyamide 6.6 (FR‐PA6.6) was prepared by the cocondensation of hexamethylene diammonium adipate (AH‐salt) with the corresponding salts of hexamethylene diamine and two different organophosphorus compounds, namely, 3‐hydroxyphenylphosphinylpropanoic acid (3‐HPP, 1) and 9,10‐dihydro‐10‐[2,3‐di (hydroxycarbonylpropyl]‐10‐phosphaphenanthrene‐10‐oxide (DDP, 2). The incorporation of the phosphorus comonomers and the thermal and physical properties of the resulting copolyamides have been studied. The phosphorus‐modified FR‐PA6.6 possesses high relative viscosities of 2.0 to 2.4, good thermal stability, and was used for the production of polyamide blends by merging FR‐PA6.6 with commercial PA6. This offered access to flame‐retarded PA6 multifilaments, which possess tensile strengths up to 0.7 GPa and elastic moduli up to 6.2 GPa. Knitted fabrics of FR‐PA6 exhibit high limiting oxygen index (LOI) values between 36 and 38 and executed burning tests demonstrate that the incorporation of phosphorus‐based comonomers improve flame retardancy significantly. The approach presented here offers a straightforward access to effective flame retardancy in nylon 6.