Characterization of Generalized Jordan Higher Derivations on Triangular rings

TL;DR

This paper proves that every additive generalized Jordan higher derivation on a certain class of triangular rings is actually a generalized higher derivation, clarifying the structure of derivations in this algebraic setting.

Contribution

It establishes that all additive generalized Jordan (triple) higher derivations on triangular rings are equivalent to generalized higher derivations, extending understanding of derivation structures.

Findings

Every additive generalized Jordan higher derivation is a generalized higher derivation.

Results apply to triangular rings formed from unital rings and bimodules.

Clarifies the structure of derivations in algebraic rings.

Abstract

Let $\mathcal A$ and $\mathcal B$ be unital rings and $\mathcal M$ be a $(\mathcal A, \mathcal B)$-bimodule, which is faithful as a left $\mathcal A$-module and also as a right $\mathcal B$-module. Let ${\mathcal U}={\rm Tri}(\mathcal A, \mathcal M, \mathcal B)$ be the associated triangular ring. It is shown that every additive generalized Jordan (triple) higher derivation on $\mathcal U$ is a generalized higher derivation.