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In this paper we develop the theory of generalized triangular matrix representation in an abstract setting. This is accomplished by introducing the concept of a set of left triangulating idempotents. These idempotents determine a generalized triangular matrix representation for an algebra. The existence of a set of left triangulating idempotents does not depend on any specific conditions on the algebras; however, if the algebra satisfies a mild finiteness condition, then such a set can be refined to a ''complete'' set of left triangulating idempotents in which each ''diagonal'' subalgebra has no nontrivial generalized triangular matrix representation. We then apply our theory to obtain new results on generalized triangular matrix representations, including extensions of several well known results. All rights of reproduction in any form reserved. i J J J J s л, then take s 0 and X s 1. J J Ä 4 Ä 4 LEMMA 3.1. Let J be a subset of 1, . . . , n and let m g 1, . . . , n _ J. Ž X X . Ä 4 Then b g S S R if and only if b Rb s 0 for each i f J j m . m l lm m m J J m m l l J J

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