Let T=Tri(A,M,B)\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {T}} = Tri ({\mathcal {A}},{\mathcal {M}},{\mathcal {B}} ) $$\end{document} be a triangular algebra where A\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {A}} $$\end{document} is a unital algebra, B\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {B}} $$\end{document} is an algebra which is not necessarily unital, and M\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {M}} $$\end{document} is a faithful (A\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {A}} $$\end{document}, B\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {B}} $$\end{document})-bimodule which is unital as a left A\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {A}} $$\end{document}-module. In this paper, under some mild conditions on T\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {T}}$$\end{document}, we show that if ϕ:T→T\documentclass[12pt]{minimal}
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\begin{document}$$ \phi : {\mathcal {T}} \rightarrow {\mathcal {T}} $$\end{document} is a linear map satisfying A,B∈T,AB=P⟹ϕ([A,B])=[A,ϕ(B)]=[ϕ(A),B],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} A,B \in {\mathcal {T}}, ~~ AB= P \Longrightarrow \phi ( [A,B])=[A,\phi (B) ]=[\phi (A) , B], \end{aligned}$$\end{document}where P is the standard idempotent of T\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {T}}$$\end{document}, then ϕ=ψ+γ\documentclass[12pt]{minimal}
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\begin{document}$$ \phi = \psi +\gamma $$\end{document} where ψ:T→T\documentclass[12pt]{minimal}
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\begin{document}$$ \psi :{\mathcal {T}} \rightarrow {\mathcal {T}}$$\end{document} is a centralizer and γ:T→Z(T)\documentclass[12pt]{minimal}
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\begin{document}$$ \gamma :{\mathcal {T}}\rightarrow Z( {\mathcal {T}}) $$\end{document} is a linear map vanishing at commutators [A, B] with AB=P\documentclass[12pt]{minimal}
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\begin{document}$$ AB=P $$\end{document} whrere Z(T)\documentclass[12pt]{minimal}
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\begin{document}$$ Z( {\mathcal {T}}) $$\end{document} is the center of T\documentclass[12pt]{minimal}
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\begin{document}$$ {\mathcal {T}}$$\end{document}. Applying our result, we characterize linear maps on T\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {T}}$$\end{document} that behave like generalized Lie 2-derivations at idempotent products as an application of above result. Our results are applied to upper triangular matrix algebras and nest algebras.
