Some Studies On Clean Rings And Rings With Stable Range Conditions

It is well known that the units and idempotents of a ring are key elements deter-mining the structure of the ring. It is proved that a ring R is a unit-regular (in the sense of J.von Neumann) ring if and only if each element of R can be written as the product of a unit and an idempotent (in any order). In parallel, if we change product to sum in this condition, that is, each element of R can be written as the sum of a unit and an idempotent, then we call R is a clean ring.This thesis does some researches on the structures and extensions of clean rings and rings with stable range conditions, obtains some new meaningful results, and consists of six chapters.In Chapter 1, the background and major results on clean rings and rings with stable range conditions are introduced. The prepare knowledge and the main results of this thesis are presented.Chapter 2 generalizes uniquely strongly clean rings to general rings. Some basic properties about uniquely strongly clean general rings are obtained. Some conditions for the upper triangular matrix ring and skew power series rings over general rings being uniquely strongly clean are given.Chapter 3 defines strongly semiclean rings. Some general properties of semiclean rings and strongly semiclean rings are given. Lastly, strongly semiclean rings of skew Hurwitz series are explored. The skew Hurwitz series of R is a semiclean ring if and only if R is.In chapter 4, replacing invertibility with quasi-invertibility in clean ring we dis-cover a new class of rings, Q-clean rings. Some basic properties of Q-clean rings are obtained. The definition of Q-clean rings with or without identity is given. According this, it is shown that the ideal of a Q-clean ring(unitial or not) is also Q-clean. Some relations between Q-clean ring and its corners are obtained.In chapter 5, some relations of quasi-unit regular and QB-rings, pseudo-unit regular and QB∞-rings are obtained. It is extended the results of Are in [6] and Chen in [35]. The sufficient and necessary conditions of an exchange ring is a QB-ring and an exchange ring is a QB∞-ring are explored.In chapter 6, rings satisfy unit weakly stable range are introduced. The basic forms of their general linear group are investigated. Similarly, Exchange rings having ideal-weakly stable range one and unit weakly (Ⅰ)-stable range rings are discussed.