Regular and generalized regular rings - Rafel, Kenneth Tamerlane Philip (Department of Mathematics and Applied Mathematics, 2016)
During his study of continuous geometries, J. von Neumann found that any complemented modular lattice satisfying certain mild conditions arises from a specific type of ring, which he termed a regular ring. In the first place, these rings turn out to be a generalization of artinian semisimple rings, and subsequent studies have shown that the concepts of regularity and some of its weaker and stronger forms are characterized by specific ideal structures which give these rings an important place in the general theory. In the first three chapters of this thesis (and in section one of chapter four), many of the known results on regularity and some of its generalizations are collected and arranged. The results are not always presented in chronological order, but are so organised as to show some development of and relationships amongst the various properties of regular and generalized regular rings. Some results which are well-known or less relevant to this development have been stated without proof. Occasionally, proofs have been supplied for results which seem only to be stated in the literature, and at times the original proofs have been modified in the interests of directness or increased generality. In all cases, the proofs of known results carry a reference in brackets, ( ). In the final two sections of the thesis, the author examines some of the properties of weakly regular rings, and defines generalizations of these rings parallel to some of those which have previously been defined for regularity. These new types of weak regularity are found to have several properties analogous to those known for the corresponding types of regularity, and in particular the ws-regular ring (defined on page 51) is characterized by an ideal structure of the same type as those found for strongly regular, regular, and weakly regular rings (as detailed on pages 53 and 54). Finally, on page 56 the author defines weak π-regularity analogously to the definition of π-regularity, and the diagram on page 59 shows how ws-regular and weakly π-regular rings fit into a pattern formed by the above mentioned known types of regularities.