An Engel condition with two generalized derivations in prime rings

Abstract

Let R be a noncommutative prime ring with extended centroid C and maximal right ring of quotients Q. Let I be a nonzero ideal of R, and let g, h be two generalized derivations of R. Suppose that ${[g(x^m)x^n-x^{s}h(x^t),x^r]}_k=0$ for all $x\in I,$ where $m,n,s,t,r,k$ are fixed positive integers. Then, one of the following conditions holds:

1. there exist $\alpha ,\beta \in C$ such that $g(x)=\alpha x$ and $h(x)=\beta x$ for all $x\in R;$

2. m = s, n = t and there exist $a\in Q$ and $\alpha \in C$ such that g(x) = xa and $h(x)=(a+\alpha )x$ for all $x\in R;$

3. C is a finite field, $R=Q\cong M_{\mathcal{\ell }}(C),$ the $\mathcal{\ell }\times \mathcal{\ell }$ matrix ring over C for some integer $\mathcal{\ell }\ge 2$ and there exist $a,u\in R$ such that g(x) = xa and h(x) = ux for all $x\in R.$

Keywords:

• Differential polynomial identity (DI)
• (generalized) derivation
• generalized polynomial identity (GPI)
• (Semi-)Prime ring

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16N60
• 16W25
• 16R50

Acknowledgement

The author is thankful to the referee for the very thorough reading of the paper and valuable suggestions.

Additional information

Funding

This research is supported by the MOST 108-2115-M-018-003 of Taiwan, R.O.C.