Triangular Matrix Representations and Triangular Matrix Extensions

Abstract

In this chapter, generalized triangular matrix representations are discussed by introducing the concept of a set of left triangulating idempotents. A criterion for a ring with a complete set of triangulating idempotents to be quasi-Baer is provided. A structure theorem for a quasi-Baer ring with a complete set of triangulating idempotents is shown using complete triangular matrix representations. A number of well known results follow as consequences of this useful structure theorem. The results which follow as a consequence include Levy’s decomposition theorem of semiprime right Goldie rings, Faith’s characterization of semiprime right FPF rings with no infinite set of central orthogonal idempotents, Gordon and Small’s characterization of piecewise domains, and Chatters’ decomposition theorem of hereditary noetherian rings. A result related to Michler’s splitting theorem for right hereditary right noetherian rings is also obtained as an application. The Baer, the quasi-Baer, the FI-extending, and the strongly FI-extending properties of (generalized) triangular matrix rings are discussed. A sheaf representation of quasi-Baer rings is obtained as an application of the results of this chapter.