On matrix methods in ring theory

Abstract

A method for representing rings as matrix rings is used to investigate the structures of several well-known classes of rings. The general method is developed in Chapter 1. In Chapter 2 nonsingular rings with essential socles are characterized by embeddings into products of full matrix rings over sfields. This generalizes the known results in the case when the rings’ identities are finite (sums of orthogonal idempotents). The results are used in Chapter 3 to study nonsingular QF-3 rings with finite identities. In particular the structure of QF-3 rings whose identities are finite and whose (principal, finitely generated) ideals are projective is determined. Chapter 4 is concerned with rings whose ideals are quasiinjective. It is shown that if such a ring is indecomposable and has more than one idempotent, then it is Artinian. The structure of these rings is then obtained. In Chapter 5 the structure of left generalized uniserial rings is determined in terms of the structure of left uniserial rings. This generalizes the known results for (left and right) generalized uniserial algebras.