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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.
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 $T$.