标题： 带有幂等元的环上的交换Jordan导子 显示英文标题

标题： Commuting Jordan derivations over a Ring with idempotents

摘要： 本文探讨了具有非平凡幂等元的素环上交换Jordan导子的行为，并证明它们成为零映射。 进一步地，它在特定条件下为素环上的交换Jordan高阶导子建立了这一结果。 事实上，它引入了环上的交换广义Jordan导子，并在某些假设下证明零映射是素环上唯一的交换广义Jordan导子。 显示英文摘要

摘要： 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.