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A note on generalized derivations as a Jordan homomorphisms
Arusha Chandrasekhar,Shailesh Kumar Tiwari 대한수학회 2020 대한수학회보 Vol.57 No.3
Let $R$ be a prime ring of characteristic different from $2$. Suppose that $F$, $G$, $H$ and $T$ are generalized derivations of $R$. Let $U$ be the Utumi quotient ring of $R$ and $C$ be the center of $U$, called the extended centroid of $R$ and let $f(x_1,\ldots,x_n)$ be a non central multilinear polynomial over $C$. If \begin{align*} &\ F(f(r_1,\ldots,r_n))G(f(r_1,\ldots,r_n))-f(r_1,\ldots,r_n)T(f(r_1,\ldots,r_n))\\ =&\ H(f(r_1,\ldots,r_n)^2) \end{align*} for all $r_1, \ldots, r_n \in R$, then we describe all possible forms of $F$, $G$, $H$ and $T$.
A NOTE ON GENERALIZED DERIVATIONS AS A JORDAN HOMOMORPHISMS
Chandrasekhar, Arusha,Tiwari, Shailesh Kumar Korean Mathematical Society 2020 대한수학회보 Vol.57 No.3
Let R be a prime ring of characteristic different from 2. Suppose that F, G, H and T are generalized derivations of R. Let U be the Utumi quotient ring of R and C be the center of U, called the extended centroid of R and let f(x<sub>1</sub>, …, x<sub>n</sub>) be a non central multilinear polynomial over C. If F(f(r<sub>1</sub>, …, r<sub>n</sub>))G(f(r<sub>1</sub>, …, r<sub>n</sub>)) - f(r<sub>1</sub>, …, r<sub>n</sub>)T(f(r<sub>1</sub>, …, r<sub>n</sub>)) = H(f(r<sub>1</sub>, …, r<sub>n</sub>)<sup>2</sup>) for all r<sub>1</sub>, …, r<sub>n</sub> ∈ R, then we describe all possible forms of F, G, H and T.
( Sarikokba Diwakar Tiwari ),( Shailesh Kumar Prasad ),( Dong Jin Kim ),( Suk Soon Choi ),( Seung-mok Lee ) 한국공업화학회 2020 공업화학 Vol.31 No.3
The role of nano bio-composites precursor to chitosan are innumerable and are known for having different applications in various branches of physical sciences. The application to the sensor development is relatively new, where only few literature works are available to address the specific and critical analysis of nanocomposites in the subject area. The bio-composites are potential and having greater affinity towards the heavy metals and several micro-pollutants hence, perhaps are having wider implications in the low or even trace level detection of the pollutants. The nano-composites could show good selectivity and suitability for the detection of the pollutants as they are found in the complex matrix. However, the greater challenges are associated using the bio-composites, since the biomaterials are prone to be oxidized or reduced at an applied potential and found to be a hinderance for the detection of target pollutants. In addition, the materials could proceed with a series of electrochemical reactions, which could produce different by-products in analytical applications, resulting in several complex phenomena in electrochemical processes. Therefore, this review addresses critically various aspects of an evaluation of nano bio-composite materials in the electrochemical detection of heavy metals and micro-pollutants from aqueous solutions.