Generalized derivations of order $2$ on multilinear polynomials in prime rings

B. Prajapati; C. Gupta

doi:10.30970/ms.58.1.26-35

B. Prajapati School of Liberal Studies Dr. B. R. Ambedkar University Delhi, INDIA https://orcid.org/0000-0001-7277-226X
C. Gupta School of Liberal Studies Dr. B. R. Ambedkar University Delhi, INDIA

DOI: https://doi.org/10.30970/ms.58.1.26-35

Keywords: Prime ring, generalized derivation, extended centroid, Martindale's quotient ring

Abstract

Let $R$ be a prime ring of characteristic different from $2$ with a right Martindale quotient ring $Q_r$ and an extended centroid $C$. Let $F$ be a non zero generalized derivation of $R$ and $S$ be the set of evaluations of a non-central valued multilinear polynomial $f(x_1,\ldots,x_n)$ over $C$. Let $p,q\in R$ be such that

$pF^2(u)u+F^2(u)uq=0$ for all $u\in S$.

Then for all $x\in R$ one of the followings holds:
1) there exists $a\in Q_r$ such that $F(x)=ax$ or $F(x)=xa$ and $a^2=0$,
2) $p=-q\in C$,
3) $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exists $a\in Q_r$ such that $F(x)=ax$ with $pa^2+a^2q=0$.

Author Biographies

B. Prajapati, School of Liberal Studies Dr. B. R. Ambedkar University Delhi, INDIA

School of Liberal Studies
Dr. B. R. Ambedkar University Delhi, INDIA

C. Gupta, School of Liberal Studies Dr. B. R. Ambedkar University Delhi, INDIA

School of Liberal Studies
Dr. B. R. Ambedkar University Delhi, INDIA

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