Given any isomorphically closed class of simple modules over a ring R analogues of the Jacobson radical and socle are studied. A triangular decomposition theorem is proved (Theorem I), in the case when R is contains no infinite family of orthogonal idempotents, and this includes an analogue of a Theorem of Gordon. We also provide in Lemma 4 a description of this socle in a triangular matrix ring when the class is the class of all simple projective modules. Finally, a structure theorem IS proved for the rings R, with no infinite family of orthogonal idempotents, in which gRg has a projective simple module for all idempotents g withgR(l-g) = 0.