Lie derivations of triangular algebras without assuming unity

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ABSTRACT In this paper we initiate the study of mapping problems on triangular algebras without assuming unity. We give the definitions of strong faithful bimodules, extreme centres, and extreme Lie derivations. Our main result is to give a description of Lie derivations on triangular algebras without assuming unity. As an application we shall give a characterization of Lie derivations on upper triangular matrix algebras over a faithful algebra.