Idempotents and structures of rings.

Recall that ann-by-ngeneralized matrix ring is defined in terms of sets of rings-bimodulesand bimodule homomorphisms, where the set of diagonal matrix unitsform a complete set of orthogonal idempotents. Moreover, an arbitrary ring with a complete set of orthogonal idempotentshas a Peirce decomposition which can be arranged into ann-by-ngeneralized matrix ringwhich is isomorphic toR. In this paper, we focus on the subclassofn-by-ngeneralized matrix rings withfor.contains all upper and all lower generalized triangular matrix rings. The triviality of the bimodule homomorphisms motivates the introduction of three new types of idempotents called the inner Peirce, outer Peirce and Peirce trivial idempotents. These idempotents are our main tools and are used to characterizeand define a new class of rings called then-Peirce rings. IfRis ann-Peirce ring, then there is a certain complete set of orthogonal idempotentssuch that. We show that everyn-by-ngeneralized matrix ringRcontains a subringSwhich is maximal with respect to being inandSis essential inRas an (S, S)-bisubmodule ofR. This allows for a useful transfer of information betweenRandS. Also, we show that any ring is either ann-Peirce ring or for eachthere is a complete set of orthogonal idempotentssuch that. Examples are provided to illustrate and delimit our results. [ABSTRACT FROM PUBLISHER]