Idempotents and structures of rings

Abstract

Recall that an n-by-n generalized matrix ring is defined in terms of sets of rings -bimodules and bimodule homomorphisms , where the set of diagonal matrix units form a complete set of orthogonal idempotents. Moreover, an arbitrary ring with a complete set of orthogonal idempotents has a Peirce decomposition which can be arranged into an n-by-n generalized matrix ring which is isomorphic to R. In this paper, we focus on the subclass of n-by-n generalized matrix rings with for . 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 characterize and define a new class of rings called the n-Peirce rings. If R is an n-Peirce ring, then there is a certain complete set of orthogonal idempotents such that . We show that every n-by-n generalized matrix ring R contains a subring S which is maximal with respect to being in and S is essential in R as an (S, S)-bisubmodule of R. This allows for a useful transfer of information between R and S. Also, we show that any ring is either an n-Peirce ring or for each there is a complete set of orthogonal idempotents such that . Examples are provided to illustrate and delimit our results.