Commuting Jordan derivations over a Ring with idempotents

Abstract

This paper explores the behaviour of commuting Jordan derivations over prime rings with non-trivial idempotents and demonstrates that they become zero maps. Further, it establishes this result for commuting Jordan higher derivations over prime rings under specific conditions. In fact, it introduces commuting generalized Jordan derivations over rings and establishes that the zero map is the only commuting generalized Jordan derivation over prime rings under certain assumptions.