Abstarct: In this paper we intruducing the concept of a set of left triangulating idempotents and investigate related condition between a ring and various ring extension. These idempotents determine a generalized tringular matix representation for a ring. Aslo we investigate the class of piecewise prime, PWP, rings which includes all piecewise domains (hence all right hereditary rings which are semiprimary or right Noetherian). For a PWP ring we determine a large class of ring extensions which have a generalized triangular matrix representation for which the diagonal rings are prime.