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- W2561771621 abstract "Let R be an associative ring with identity 1 ≠ 0. An element a ∈ R is called cleanif there exists an idempotent e and a unit u in R such that a = e + u, and a is calledstrongly 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 givenby Nicholson in 1977 in a study of exchange rings and that of a strongly clean ring wasintroduced also by Nicholson in 1999 as a natural generalization of strongly π-regularrings. Besides strongly π-regular rings, local rings give another family of strongly cleanrings.The main part of this thesis deals with the question of when a matrix ring is stronglyclean. This is motivated by a counter-example discovered by Sanchez Campos and Wang-Chenrespectively to a question of Nicholson whether a matrix ring over a strongly cleanring is again strongly clean. They both proved that the 2 x 2 matrix ring M₂(Z₍₂₎) is notstrongly clean, where Z₍₂₎ is the localization of Z at the prime ideal (2). The followingresults 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) forwhich 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 bestrongly clean.• Necessary and sufficient conditions for Mn(R) over a commutative local ring R tobe strongly clean.• A family of strongly clean triangular matrix rings.• New families of strongly π-regular (of course strongly clean) matrix rings over noncommutativelocal rings or strongly π-regular rings.Another part of this thesis is about the so-called g(x)-clean rings. Let C(R) be thecenter of R and let g(x) be a polynomial in C(R)[x]. An element a ∈ R is called g(x)cleanif a == e + u where g(e) == 0 and u is a unit of R. The ring R is g(x)-clean ifevery 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. Therelationship between clean rings and g(x)-clean rings is discussed here." @default.
- W2561771621 created "2017-01-06" @default.
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- W2561771621 date "2007-08-01" @default.
- W2561771621 modified "2023-09-27" @default.
- W2561771621 title "Strongly clean rings and g(x)-clean rings" @default.
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