Classes of strongly clean rings

Abstract

A ring is called clean if every element can be written as the sum of an idempotent and a unit, and a ring is further called strongly clean if every element can be written as the sum of an idempotent and a unit that commute. The aim of this work is to study strongly clean rings by representing them as full endomorphism rings of certain modules. We are motivated by several fundamental questions that have been raised concerning strongly clean rings. We begin by placing the characterization of a strongly clean endomorphism in a more general context and then use these ideas to develop certain new classes of strongly clean rings. In Chapter 2, we investigate upper-triangular matrix rings over local rings. We show that Tn(R) is strongly clean for a large class of local rings R. We then use a skew power series ring construction to produce a local ring A such that the (semiperfect) ring T2 (A) is not strongly clean. In Chapter 3, we develop the class of strongly nil clean rings, a natural sub-class of the strongly 7r-regular rings. We show, among other things, that Tn(R) is strongly nil clean for any strongly nil clean local ring R, and we completely characterize the class of uniquely nil clean rings. In Chapter 4 we then analyze the class of strongly 7-rad clean rings with a

view toward studying certain open questions in the theory of strongly clean rings. Specifically, we answer several questions that are raised in Chapter 2, and we additionally offer some perspective on the question of whether every unit regular ring is strongly clean and on the question of whether all strongly clean rings have stable range 1.