On b-generalized skew derivations having centralizing commutator in prime rings

Abstract

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, $f(x_1,\dots ,x_n)$ be a multilinear polynomial over C, which is not central valued on R. Suppose $P\in R$ and G is a non-zero b-generalized skew derivation of R such that $[P,G(u)u]\in C$ for all $u\in f(R).$ Then one of the following holds:

1. $P\in C,$

2. $f{(x_1,\dots ,x_n)}^2$ is central valued and there exists $Q\in U$ such that $G(x)=Qx$ for all $x\in R$ and $[P,Q]\in C.$

We apply this result to investigate the commutator $[F(v)v,G(u)u]\in C$ for all $u,v\in f(R)$ where F and G are b-generalized skew derivations on R.

Keywords:

• Prime ring
• Utumi quotient ring
• extended centroid
• b-generalized skew derivation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16W25

Acknowledgment

The authors are highly thankful to the referee for his/her several useful suggestions.

Additional information

Funding

B. Prajapati is supported by the research grant DST-SERB EMR/2016/001550.