A Study On Direct Summand Submodules Over Noncommutative Rings

Abstract

A natural problem to consider in ring and module theory is to investigate the cancellation property of a given object. This problem was first considered by Jónsson and Tarski for any algebraic system and then gave rise to many variations related to the cancellation theme such as substitution and internal cancellation. In the mid-30s of the last century, just before the cancellation problem was treated for any algebraic system by Jónsson and Tarski, a ground-breaking invention was made by von Neumann. He developed the theory of continuous geometries. One of the main ideas of this new structure was the construction of a dimension function whose range is a continuum of real numbers and this construction was based on the perspectivity relation. Throughout this work, we discuss new concepts derived from cancellation and continuity. This dissertation consists of four chapters. In the first chapter, we recall the ring and module theoretical properties that play an important role within our framework like stable range conditions, the exchange property, and perspectivity. In the second chapter, we study the class of internally cancellable rings, i.e., the class of rings that satisfy internal cancellation property with respect to their one-sided ideals. By considering a condition, we obtain new characterizations of internally cancellable rings, unit regular rings, and rings with stable range one. We also investigate internally cancellable rings with the summand sum property. In Chapter 3, we introduce the lifting of elements having (idempotent) stable range one from a quotient of a ring $R$ modulo a two-sided ideal $I$ by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring $R$ is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal iff $R$ is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements. In the last chapter, we study the most recent variations of continuity and discreteness concepts, namely $C4$- and $D4$-modules, in terms of perspective direct summands by providing new characterizations and results. Endomorphism rings of $C4$-modules and extensions of right $C4$-rings are also investigated. Decompositions of $C4$-modules with restricted ACC on direct summands and $D4$-modules with restricted DCC on direct summands are obtained.