Department of Mathematics,
Faculty of Science,
Islamic University of Madinah,
Saudi Arabia.
E-mail: ansari.abuzaid@gmail.com

Abstract:

Let S be a (p+m)!-torsion-free semiprime ring, and let B, k : S→S be additive mappings satisfying, for all y ∈ S, the identity

where γ and θ are automorphisms of S. Then B is a generalized (γ,θ)-derivation on B, with k serving as its associated (γ,θ)-derivation. Moreover, analogously, we establish an additional result demonstrating that, if two additive mappings satisfy a prescribed identity, then each such mapping constitutes a generalized left (γ,θ)-derivative having left (γ,θ)-derivative. Furthermore, the appropriateness of this hypothesis is substantiated by explicit illustrative examples.

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