Jordan homoderivation behavior of generalized derivations in prime rings

Nripendu Bera; Basudeb Dhara

doi:10.3842/umzh.v75i9.7241

Nripendu Bera Department of Mathematics, Jadavpur University, Kolkata, India
Basudeb Dhara Department of Mathematics, Belda College, Belda, Paschim Medinipur, India

DOI: https://doi.org/10.3842/umzh.v75i9.7241

Keywords: Generalized derivation, Homoderivation, Utumi quotient ring, Extended centroid, Multilinear polynomial, Prime ring

Abstract

UDC 512.5

Suppose that $R$ is a prime ring with ${\rm char}(R)\neq 2$ and $f(\xi_1,\ldots,\xi_n)$ is a noncentral multilinear polynomial over $C( = Z(U)),$ where $U$ is the Utumi quotient ring of $R.$ An additive mapping $h\colon R\rightarrow R$ is called homoderivation if $h(ab) = h(a)h(b)+h(a)b+ah(b)$ for all $a,b\in R.$ We investigate the behavior of three generalized derivations $F,$ $G,$ and $H$ of $R$ satisfying the condition $$F(\xi^2) = G(\xi)^2+H(\xi)\xi+\xi H(\xi)$$ for all $\xi \in f(R) = \{f(\xi_1,\ldots,\xi_n) \mid \xi_1,\ldots,\xi_n\in R\}.$

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