Commuting Jordan derivations over a Ring with idempotents

TL;DR

This paper investigates commuting Jordan derivations and higher derivations over prime rings with idempotents, proving they are zero maps under certain conditions, and introduces generalized Jordan derivations with similar results.

Contribution

It establishes that commuting Jordan derivations and higher derivations over prime rings are zero maps, and introduces commuting generalized Jordan derivations with similar properties.

Findings

Commuting Jordan derivations over prime rings are zero maps.

Commuting Jordan higher derivations over prime rings are zero maps under certain conditions.

Zero map is the only commuting generalized Jordan derivation over prime rings under specific assumptions.

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.