In this thesis by defining orthogonal decomposition for modules, we prove that an R-module M has only finitely many fully invariant direct summand if and only if End(M) has triangulating dimension n. It is shown that triangulating dimension of R-module M is Morita invariant and when R is an Artinian pir ring, triangulating dimension of M is the number of socle components of M. If R is commutative then R is perfect (resp. a finite direct product of domains) if and only if it is semi Artinian (resp. semiprime extending) with finite triangulating dimension. Also we define piecewise semiprime (PWSP) rings R in terms of a set of left triangulating idempotents in R. The dir=ltr
