Some generalized Jordan maps on triangular rings force additivity

TL;DR

This paper proves that certain generalized Jordan maps on triangular rings are necessarily additive, extending the understanding of structure-preserving maps in algebraic settings.

Contribution

It establishes additivity of generalized Jordan maps and related maps on triangular rings under specific algebraic conditions.

Findings

Generalized Jordan maps satisfying specific conditions are additive.

Maps satisfying $T(ab)=T(a)b=aT(b)$ are additive.

Maps satisfying a certain functional equation are additive.

Abstract

In this paper, we show that a map $\delta$ over a triangular ring $\mathcal{T}$ satisfying $\delta(ab+ba)=\delta(a)b+a \tau(b)+\delta(b)a+b\tau(a)$, for all $a,b\in \mathcal{T}$ and for some maps $\tau$ over $\mathcal{T}$ satisfying $\tau(ab+ba)=\tau(a)b+a \tau(b)+\tau(b)a+b\tau(a)$, is additive. Also, it is shown that a map $T$ on $\mathcal{T}$ satisfying $T(ab)=T(a)b=aT(b)$, for all $a,b\in \mathcal{T}$, is additive. Further, we establish that if a map $D$ over $\mathcal{T}$ satisfies $(m+n)D(ab)=2mD(a)b+2naD(b)$, for all $a,b\in \mathcal{T}$ and integers $m,n\geq 1$, then $D$ is additive.