Multiplicative generalized Lie n-derivations of unital rings with idempotents

Let A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {A}$$\end{document} be a unital ring with a nontrivial idempotent. In this paper, it is shown that under certain conditions every multiplicative generalized Lie n-derivation Δ:A→A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta :\mathfrak {A}\rightarrow \mathfrak {A}$$\end{document} is of the form Δ(u)=zu+δ(u),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (u)=zu+\delta (u),$$\end{document} where z∈Z(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in \mathcal {Z}(\mathfrak {A})$$\end{document} and δ:A→A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta :\mathfrak {A}\rightarrow \mathfrak {A}$$\end{document} is a multiplicative Lie n-derivation. The main result is then applied to some classical examples of unital rings with nontrivial idempotents such as triangular rings, matrix rings, nest algebras, and algebras of bounded linear operators.