Strongly clean rings and g(x)-clean rings

Abstract

Let R be an associative ring with identity 1 ≠ 0. An element a ∈ R is called clean if there exists an idempotent e and a unit u in R such that a = e + u, and a is called strongly clean if, in addition, eu = ue. The ring R is called clean (resp., strongly clean) if every element of R is clean (resp., strongly clean). The notion of a clean ring was given by Nicholson in 1977 in a study of exchange rings and that of a strongly clean ring was introduced also by Nicholson in 1999 as a natural generalization of strongly π-regular rings. Besides strongly π-regular rings, local rings give another family of strongly clean rings. The main part of this thesis deals with the question of when a matrix ring is strongly clean. This is motivated by a counter-example discovered by Sanchez Campos and Wang-Chen respectively to a question of Nicholson whether a matrix ring over a strongly clean ring is again strongly clean. They both proved that the 2 x 2 matrix ring M₂(Z₍₂₎) is not strongly clean, where Z₍₂₎ is the localization of Z at the prime ideal (2). The following results are obtained regarding this question: • Various examples of non-strongly clean matrix rings over strongly clean rings. • Completely determining the local rings R (commutative or noncommutative) for which M₂ (R) is strongly clean. • A necessary condition for M₂(R) over an arbitrary ring R to be strongly clean. • A criterion for a single matrix in Mn(R) to be strongly clean when R has IBN and every finitely generated projective R-module is free. • A sufficient condition for the matrix ring Mn(R) over a commutative ring R to be strongly clean. • Necessary and sufficient conditions for Mn(R) over a commutative local ring R to be strongly clean. • A family of strongly clean triangular matrix rings. • New families of strongly π-regular (of course strongly clean) matrix rings over noncommutative local rings or strongly π-regular rings. Another part of this thesis is about the so-called g(x)-clean rings. Let C(R) be the center of R and let g(x) be a polynomial in C(R)[x]. An element a ∈ R is called g(x)clean if a == e + u where g(e) == 0 and u is a unit of R. The ring R is g(x)-clean if every element of R is g(x)-clean. The (x² - x )-clean rings are precisely the clean rings. The notation of a g(x)-clean ring was introduced by Camillo and Simon in 2002. The relationship between clean rings and g(x)-clean rings is discussed here.