Additivity of multiplicative generalized Jordan maps on triangular rings

Abstract

This paper presents three different conditions for the additivity of a map on a triangular ring $\mathcal{T}$. First, we prove a map $\delta$ on $\mathcal{T}$ satisfying $delta(a_1b_1+b_1a_1)=\delta(a_1)b_1 +a_1 \tau(b_1)+\delta(b_1)a_1 + b_1\tau(a_1)$ for all $a_1,b_1\in \mathcal{T}$ and for some maps $\tau$ over $\mathcal{T}$ satisfying $\tau(a_1b_1+b_1a_1)=\tau(a_1)b_1+a_1 \tau(b_1)+\tau(b_1)a_1+b_1\tau(a_1)$, is additive. Secondly, it is shown that a map $T$ on $\mathcal{T}$ satisfying $T(a_1b_1)=T(a_1)b_1=a_1T(b_1)$ for all $a_1,b_1\in \mathcal{T}$ is additive. Finally, we show that if a map $D$ over $\mathcal{T}$ satisfies $(m+n)D(a_1b_1)=2mD(a_1)b_1+2na_1D(b_1)$ for all $a_1,b_1\in \mathcal{T}$ and integers $m,n\geq 1$, then $D$ is additive.

Keywords

• Additivity
• Jordan derivation
• two-sided centralizer
• $(m
• n)$-derivation
• triangular ring

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