Annihilating and co-commuting conditions with generalized skew derivations

Abstract

Let R be a prime ring of characteristic different from 2, Q_r its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R associated with the same automorphism α of R, I a non-central ideal of R and $0\ne a\in R$. If $a(F(x)x-xG(x))=0$, for all $x\in I$, then there exist $b,b\prime ,q\in Q_r$ and $\lambda ,\vartheta \in C$ such that $F(x)=bx+\mathit{qxb}\prime ,G(x)=(\vartheta b\prime +\lambda )x$, for any $x\in R$, with $a(b-\lambda )=0$ and $a(q-\vartheta )=0$. In particular, if α is not an inner automorphism of R, then F(x) = bx, $G(x)=\lambda x$, for any $x\in R$, with $a(b-\lambda )=0$.

KEYWORDS:

• Generalized skew derivation
• prime ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W25
• 16N60