Pair of generalized derivations acting on multilinear polynomials in prime rings

Authors : Basudeb DHARA, Sukhendu KAR, Priyadwip DAS

Pages : 740-753

Doi:10.15672/hujms.588747

View : 25 | Download : 10

Publication Date : 2020-04-02

Article Type : Research Paper

Abstract :Let $R$ be a noncommutative prime ring of characteristic different from $2$ with Utumi quotient ring $U$ and extended centroid $C$ and $finsert ignore into journalissuearticles values(r_1,\ldots,r_n);$ be a multilinear polynomial over $C$, which is not central valued on $R$. Suppose that $F$ and $G$ are two nonzero generalized derivations of $R$ such that $G\neq Id$ insert ignore into journalissuearticles values(identity map); and $$Finsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(r);^2);=Finsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(r););Ginsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(r););+Ginsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(r););Finsert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(r););$$ for all $r=insert ignore into journalissuearticles values(r_1,\ldots,r_n);\in R^n$. Then one of the following holds: insert ignore into journalissuearticles values(1); there exist $\lambda \in C$ and $\mu \in C$ such that $Finsert ignore into journalissuearticles values(x);=\lambda x$ and $Ginsert ignore into journalissuearticles values(x);=\mu x$ for all $x\in R$ with $2\mu=1$; insert ignore into journalissuearticles values(2); there exist $\lambda \in C$ and $p,q\in U$ such that $Finsert ignore into journalissuearticles values(x);=\lambda x$ and $Ginsert ignore into journalissuearticles values(x);=px+xq$ for all $x\in R$ with $p+q\in C$, $2insert ignore into journalissuearticles values(p+q);=1$ and $finsert ignore into journalissuearticles values(x_1,\ldots,x_n);^2$ is central valued on $R$; insert ignore into journalissuearticles values(3); there exist $\lambda \in C$ and $a\in U$ such that $Finsert ignore into journalissuearticles values(x);=[a,x]$ and $Ginsert ignore into journalissuearticles values(x);=\lambda x$ for all $x\in R$ with $finsert ignore into journalissuearticles values(x_1,\ldots,x_n);^2$ is central valued on $R$; insert ignore into journalissuearticles values(4); there exist $\lambda \in C$ and $a,b\in U$ such that $Finsert ignore into journalissuearticles values(x);=ax+xb$ and $Ginsert ignore into journalissuearticles values(x);=\lambda x$ for all $x\in R$ with $a+b\in C$, $2\lambda =1$ and $finsert ignore into journalissuearticles values(x_1,\ldots,x_n);^2$ is central valued on $R$; insert ignore into journalissuearticles values(5); there exist $a, p\in U$ such that $Finsert ignore into journalissuearticles values(x);=xa$ and $Ginsert ignore into journalissuearticles values(x);=px$ for all $x\in R$, with $insert ignore into journalissuearticles values(p-1);a=-ap\in C$ and $finsert ignore into journalissuearticles values(x_1,\ldots,x_n);^2$ is central valued on $R$; insert ignore into journalissuearticles values(6); there exist $a, q\in U$ such that $Finsert ignore into journalissuearticles values(x);=ax$ and $Ginsert ignore into journalissuearticles values(x);=xq$ for all $x\in R$ with $ainsert ignore into journalissuearticles values(q-1);=-qa\in C$ and $finsert ignore into journalissuearticles values(x_1,\ldots,x_n);^2$ is central valued on $R$.
Keywords : prime ring, derivation, generalized derivation, extended centroid