There are no nontrivial two-sided multiplicative (generalized)-skew derivations in prime rings

TL;DR

This paper proves that in prime rings, multiplicative (generalized)-skew derivations are essentially either derivations or skew derivations, showing no nontrivial two-sided multiplicative (generalized)-skew derivations exist.

Contribution

It demonstrates that the simultaneous satisfaction of defining identities restricts multiplicative (generalized)-skew derivations to known types in prime rings.

Findings

No nontrivial two-sided multiplicative (generalized)-skew derivations exist in prime rings.

Such derivations are either multiplicative derivations or generalized skew derivations.

Only one identity is needed to define new classes of derivations in prime rings.

Abstract

As originally defined by Ashraf and Mozumder, multiplicative (generalized)-skew derivations must satisfy two identities. In this short note we show that, as a consequence of the simultaneous satisfaction of both identities, a multiplicative (generalized)-skew derivation of a prime ring is either a multiplicative (generalized) derivation (i.e., not skew), or a generalized skew derivation (i.e., additive). Therefore only one of the identities should be taken in the definition of multiplicative (generalized)-skew derivations in order to get a new class of derivations in prime rings.