Characterization of $n$-Lie $(m_1,\cdots ,m_n)$-derivations on triangular rings

The central objective of this paper was to investigate the structure of $n$-Lie $(m_1,\dots ,m_n)$-derivations on triangular rings from two distinct perspectives: the faithful bimodule property and the maximal left ideal quotient ring. Under certain assumptions, we established that every $2$-Lie $(m_1,m_2)$-derivation decomposes into the sum of an inner derivation, an extremal biderivation, and a bilinear central mapping. Furthermore, using mathematical induction, we proved that for $n\ge 3$, each $n$-Lie $(m_1,\dots ,m_n)$-derivation can be expressed as the sum of an extremal $n$-derivation and an $n$-linear central mapping. Notably, our analysis revealed that these structural forms remain invariant regardless of whether we adopt the faithful bimodule approach or the maximal left ideal quotient ring framework. As significant corollaries, we obtained explicit structural descriptions of $n$-Lie $(m_1,\dots ,m_n)$-derivations on upper triangular rings and nest algebras. Additionally, our investigation yielded several interesting extensions and generalizations of existing results in this domain.