Certain functional identities involving a pair of homogeneous derivations with central values in gr-prime rings

TL;DR

This paper investigates functional identities involving homogeneous derivations in gr-prime rings, establishing conditions under which the rings are commutative and highlighting limitations in extending these results to gr-semiprime rings.

Contribution

It introduces new commutativity conditions for gr-prime rings involving pairs of homogeneous derivations, extending classical prime ring results to the graded setting.

Findings

Under certain conditions, gr-prime rings are commutative.

Results do not extend to gr-semiprime rings.

Provides new insights into derivations in graded ring theory.

Abstract

In this paper, we explore functional identities with central values in gr-prime rings involving pairs of homogeneous derivations. We establish commutativity conditions that extend classical results from prime rings to the graded setting. In particular, we show that under certain conditions on homogeneous derivations, the ring must be commutative. Furthermore, we demonstrate that these results cannot be extended to gr-semiprime rings.