Idempotents and structures of rings

Pham Ngoc, Anh and Birkenmeier, G. F. and van Wyk, Leon (2016) Idempotents and structures of rings. LINEAR AND MULTILINEAR ALGEBRA, 64 (10). pp. 2002-2029. ISSN 0308-1087

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Abstract

Recall that an n-by-n generalized matrix ring is defined in terms of sets of rings (Formula presented.)-bimodules (Formula presented.) and bimodule homomorphisms (Formula presented.), where the set of diagonal matrix units (Formula presented.) form a complete set of orthogonal idempotents. Moreover, an arbitrary ring with a complete set of orthogonal idempotents (Formula presented.) has a Peirce decomposition which can be arranged into an n-by-n generalized matrix ring (Formula presented.) which is isomorphic to R. In this paper, we focus on the subclass (Formula presented.) of n-by-n generalized matrix rings with (Formula presented.) for (Formula presented.). (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) such that (Formula presented.). We show that every n-by-n generalized matrix ring R contains a subring S which is maximal with respect to being in (Formula presented.) 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 (Formula presented.) there is a complete set of orthogonal idempotents (Formula presented.) such that (Formula presented.). Examples are provided to illustrate and delimit our results. © 2016 Taylor & Francis

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