Clean Rings And Their Generalizations

This dissertation focuses on the development of clean rings and their general-izations. We study potent rings, f-clean rings and strong cleanness of matrix ringsover noncommutative local rings. Moreover, we study lifting-ideals in associativepairs.It is well known that the units and idempotents of a ring are key elementsdetermining the structure of the ring. It is proved that a ring R is a unit-regular(inthe sense of J. von Neumann) ring if and only if each element of R can be writtenas the product of a unit and an idempotent(in any order). In parallel, if we changeproduct to sum in this condition; that is, each element can be written as the sum ofa unit and an idempotent, then we call R is a clean ring. This dissertation consistsof five chapters, whose main contents are described as follows:In the preface, some background materials for clean rings and some related ringsand their development are introduced. Also we review some of the general facts onassociative pairs.Chapter 2 studies the potent rings and their various properties. We investigatethe structure of potent rings, and show that if R is a potent ring whose idempotentsare trivial, then R is precisely a local ring(Theorem 2.3.6).In recent years, many authors have been studying the topic of determiningwhen Mn(R) is strongly clean for a local ring R. But all of them were constraint incommutative local rings. In Chapter 3, we focus on the question of when M2(R) isstrong clean for a noncommutative local ring R. We establish a criteria for M2(R)to be a strongly clean ring(Theorem 3.2.7). And as examples, we presented a newfamily of strongly clean rings(Theorem 3.3.2).Chapter 4 is devoted to studying two new classes of rings: f-clean rings andrings having many full elements. We examine various properties of f-clean ringsand give some examples of such rings. Also, we investigate the characterizations ofelements in rings with many full elements. Let C = (A, B, V, W,ψ,φ) be theMorita Context(see the definition in Chapter 4). We prove that A, B have many fullelements, so have Mn(A) for any n≥1 and the Morita Context ring C (Theorem4.3.18 and Theorem 4.3.21, respectively).In Chapter 5, we are intended to extend the concept of rings in which idem-potents lift in R modulo J(R) and their properties to the context of associative pairs. We prove that idempotents lift modulo Rad A if and only if idempotents liftmodulo each left ideal contained in Rad A(Theorem 5.2.6). Moreover, we establisha relationship between associative pair A and its standard embedding UA on theproperty of lifting idempotents modulo the Jacobson radical(Theorem 5.2.11).Finally, we review our dissertation and discuss some further questions whichwe are interested in.