Strongly clean rings and g(x)-clean rings

Let R be an associative ring with identity 1 ≠ 0. An element a ∈ R is called clean if there exists an idempotent e and a unit u in R such that a = e + u, and a is called strongly clean if, in addition, eu = ue. The ring R is called clean (resp., strongly clean) if every element of R is clean (resp., strongly clean). The notion of a clean ring was given by Nicholson in 1977 in a study of exchange rings and that of a strongly clean ring was introduced also by Nicholson in 1999 as a natural generalization of strongly π-regular rings. Besides strongly π-regular rings, local rings give another family of strongly clean rings. The main part of this thesis deals with the question of when a matrix ring is strongly clean. This is motivated by a counter-example discovered by Sanchez Campos and Wang-Chen respectively to a question of Nicholson whether a matrix ring over a strongly clean ring is again strongly clean. They both proved that the 2 x 2 matrix ring M₂(Z₍₂₎) is not strongly clean, where Z₍₂₎ is the localization of Z at the prime ideal (2). The following results are obtained regarding this question: • Various examples of non-strongly clean matrix rings over strongly clean rings. • Completely determining the local rings R (commutative or noncommutative) for which M₂ (R) is strongly clean. • A necessary condition for M₂(R) over an arbitrary ring R to be strongly clean. • A criterion for a single matrix in Mn(R) to be strongly clean when R has IBN and every finitely generated projective R-module is free. • A sufficient condition for the matrix ring Mn(R) over a commutative ring R to be strongly clean. • Necessary and sufficient conditions for Mn(R) over a commutative local ring R to be strongly clean. • A family of strongly clean triangular matrix rings. • New families of strongly π-regular (of course strongly clean) matrix rings over noncommutative local rings or strongly π-regular rings. Another part of this thesis is about the so-called g(x)-clean rings. Let C(R) be the center of R and let g(x) be a polynomial in C(R)[x]. An element a ∈ R is called g(x)clean if a == e + u where g(e) == 0 and u is a unit of R. The ring R is g(x)-clean if every element of R is g(x)-clean. The (x² - x )-clean rings are precisely the clean rings. The notation of a g(x)-clean ring was introduced by Camillo and Simon in 2002. The relationship between clean rings and g(x)-clean rings is discussed here

Optimization of Extraction Process of Polysaccharide from Sophora japonica by Compound Enzyme Method and Its Antioxidant Activity

Objective: Sophora japonica polysaccharides were extracted by compound enzyme method, and the extraction process was optimized. The antioxidant activity in vitro was evaluated. Methods: The effects of addition amount of compound enzyme, pH, proportion of compound enzyme and enzymatic hydrolysis time on the extraction yield were investigated by single factor experiment. On the basic of single factor experiment, response surface method was used to determine the optimal extraction parameters of Sophora japonica polysaccharide. Compared with VC, the antioxidant activity of Sophora japonica polysaccharides was investigated by measuring the scavenging rate of DPPH· and ABTS+· and the total reducing power. Results: The optimal extraction parameters of Sophora japonica polysaccharides were as follows: The addition amount of compound enzyme was 23.8 mg/g, pH4.8, and the ratio of pectinase to cellulase was 0.912:1. Under this process, the yield of Sophora japonica polysaccharides was 10.71%, and the extracted polysaccharide showed good scavenging ability for DPPH· and ABTS+·. When the concentration of the polysaccharide solution was 2.8 mg/mL, the scavenging rate of DPPH· and ABTS+· reached 94.19% and 99.79% of VC at the same concentration, respectively, and the total reducing power reached 75.99% of VC. Conclution: Sophora japonica polysaccharide could be effectively extracted by compound enzymatic method and its antioxidant activity could be improved, which provided a theoretical reference for the development of functional food of Sophora japonica polysaccharide

Arrangement models of alkylammonium cations in the interlayer of HDTMA+ pillared montmorillonites

The orientation of HDTMA+ in the interlayer of organic pillared montmorillonites prepared at different concentrations of HDTMA+ have been studied using X-ray powder diffraction (XRD) and theoretical calculation. A series of arrangement models of HDTMA+ in the interlayer of montmorillonite have been proposed as lateral-monolayer (LM), lateral-bilayer (LB), pseudotrilayer (PT), paraffin-type-monolayer (PM), paraffin-type-bilayer (PB). With the increase of the concentration of HDTMA+, the arrangement model of HDTMA+ in the interlayer of montmorillonites changes as lateral-monolayer→lateral-bilayer→paraffin-type monolayer→pseudotrilayer→paraffin-type-bilayer and the packing density of HDTMA+ in the interlayer increases gradually. In the intermediate stages, different models may coexist

Some families of strongly clean rings

AbstractA ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer n⩾2, the paper deals with the question whether the strongly clean property of Mn(R[[x]]), MnR[x](xk), and Mn(RC2) follows from the strongly clean property of Mn(R). This is ‘Yes’ if n=2 by a known result