Idempotents and structures of rings

We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular generalized matrix rings are pertinent examples of the class of rings which we study. The triviality of the particular bimodule homomorphisms motivates the introduction of three new types of idempotents, namely inner Peirce trivial idempotents, outer Peirce trivial idempotents and Peirce trivial idempotents. These idempotents provide the main tools in our investigations.