Multiplicative generalized Jordan $n$-derivations of unital rings with idempotents

TL;DR

This paper proves that under certain conditions, multiplicative generalized Jordan n-derivations on unital rings with idempotents are additive and can be expressed as a sum of a central element times the argument plus a Jordan n-derivation, with applications to various classical rings.

Contribution

It establishes the additivity and explicit form of multiplicative generalized Jordan n-derivations on unital rings with idempotents, extending understanding in ring theory.

Findings

Every such derivation is additive under certain conditions.

The derivation can be expressed as a sum of a central element times the input and a Jordan n-derivation.

Applications to classical rings like matrix rings and von Neumann algebras demonstrate the results' broad relevance.

Abstract

Let $\mathfrak{A}$ be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Jordan $n$-derivation $\Delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is additive. More precisely, it is proved that $\Delta$ is of the form $\Delta(t)=\mu t+\delta(t),$ where $\mu\in\mathcal{Z}(\mathfrak{A})$ and $\delta:\mathfrak{A}\rightarrow\mathfrak{A}$ is a Jordan $n$-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, prime rings, nest algebras, standard operator algebras, and von Neumann algebras.