On triangular $\mathscr{n}$-matrix rings having multiplicative Lie type derivations

Abstract

Let $1<\mathscr{n}\in \mathbb{Z}^+$ and $\mathscr{T}$ be a triangular $\mathscr{n}$-matrix ring. This manuscript reveals  that under a few moderate presumptions, a map $\mathscr{L} : \mathscr{T} \rightarrow \mathscr{T}$ could be a multiplicative Lie $\texttt{N}$-derivation iff $\mathscr{L}(\mathscr{X}) = \mathscr{d}(\mathscr{X}) + \zeta(\mathscr{X})$ holds on every $\mathscr{X} \in \mathscr{T},$ where $\mathscr{d} : \mathscr{T} \rightarrow \mathscr{T}$ is an additive derivation and $\zeta :\mathscr{T} \rightarrow \mathscr{Z}(\mathscr{T})$ is a central valued map that disappears on all Lie $\texttt{N}$-products.