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Let $R$ be a noncommutative prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C$ the extended centroid of $R$ and $f(x_1,\ldots,x_n)$ a noncentral multilinear polynomial over $C$ in $n$ noncommuting variables. Denote by $f(R)$ the set of all the evaluations of $f(x_1,\ldots,x_n)$ on $R$. If $d$ is a nonzero derivation of $R$ and $G$ a nonzero generalized derivation of $R$ such that $$ d(G(u)u)\in Z(R)$$ for all $u \in f(R)$, then $f(x_1,\ldots,x_n)^2$ is central-valued on $R$ and there exists $b\in U$ such that $G(x) = bx$ for all $x\in R$ with $d(b) \in C$. As an application of this result, we investigate the commutator $[F(u)u$, $G(v)v]\in Z(R)$ for all $u, v\in f(R)$, where $F$ and $G$ are two nonzero generalized derivations of $R$.
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참고문헌 (13)
참고문헌 신청
A. Ali / 2014 / Commuting values of generalized derivations on multilinear polynomials / Comm. Algebra 42(9):3699~3707
C.-M. Chang / 1998 / Annihilators of power values of derivations in prime rings / Comm. Algebra 26(7):2091~2113
C.-L. Chuang / 1987 / The additive subgroup generated by a polynomial / Israel J. Math 59(1):98~106
C. -L. Chuang / 1988 / GPIs having coecients in Utumi quotient rings / Proc. Amer. Math. Soc 103(3):723~728
V. De Filippis / 2014 / Cocentralizing generalized derivations on multilinear poly-nomial on right ideals of prime rings / Demonstr. Math 47(1):22~36
C.-M. Chang / 1998 / Annihilators of power values of derivations in prime rings / Comm. Algebra 26(7):2091~2113
C.-L. Chuang / 1987 / The additive subgroup generated by a polynomial / Israel J. Math 59(1):98~106
C. -L. Chuang / 1988 / GPIs having coecients in Utumi quotient rings / Proc. Amer. Math. Soc 103(3):723~728
V. De Filippis / 2014 / Cocentralizing generalized derivations on multilinear poly-nomial on right ideals of prime rings / Demonstr. Math 47(1):22~36
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