Some generalized Jordan maps on triangular rings force additivity

In this paper, we show that a map $δ$ over a triangular ring $\mathcal{T}$ satisfying $δ(ab+ba)=δ(a)b+a τ(b)+δ(b)a+bτ(a)$, for all $a,b\in \mathcal{T}$ and for some maps $τ$ over $\mathcal{T}$ satisfying $τ(ab+ba)=τ(a)b+a τ(b)+τ(b)a+bτ(a)$, 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.

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