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 pi-regular rings. Besides strongly pi-regular rings, local rings give another family of strongly clean 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 (x 2 -- 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.;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&parl0;Z 2&parr0; is not strongly clean, where Z2 is the localization of Z at the prime ideal (2). The following results are obtained regarding this question: (1) Various examples of non-strongly clean matrix rings over strongly clean rings. (2) Completely determining the local rings R (commutative or noncommutative) for which M2 (R) is strongly clean. (3) A necessary condition for M2 (R) over an arbitrary ring R to be strongly clean. (4) 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. (5) A sufficient condition for the matrix ring Mn (R) over a commutative ring R to be strongly clean. (6) Necessary and sufficient conditions for Mn (R) over a commutative local ring R to be strongly clean. (7) A family of strongly clean triangular matrix rings. (8) New families of strongly pi-regular (of course strongly clean) matrix rings over noncommutative local rings or strongly pi-regular rings.
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机译：令R为标识为1≠0的缔合环。如果元素a∈R在R中存在一个等幂e和一个单位u，使得a = e + u，则称其为clean；如果另外存在a，则称其为强清洗。 ，eu = ue。如果R的每个元素都是干净的（则是非常干净的），则将环R称为干净的（即是非常干净的）。 Nicholson在1977年对交换环的研究中提出了清洁环的概念，Nicholson也在1999年引入了强清洁环的概念，作为强pi正则环的自然概括。除了强π正则环，局部环还提供了另一组强清洁环。本论文的另一部分是关于所谓的g（x）-清洁环。令C（R）为R的中心，令g（x）为C（R）[x]中的多项式。如果a = e + u其中g（e）= 0且u是R的单位，则元素∈R称为g（x）-clean。如果R的每个元素均为R，则环R为g（x）-clean。 g（x）-干净。 （x 2-x）清洁环正是清洁环。 Camillo和Simon在2002年引入了ag（x）-清洁环的概念。这里讨论了清洁环与g（x）-清洁环的关系。矩阵环非常干净。这是由Sanchez Campos和Wang-Chen分别发现的反例引起的，该反例是针对Nicholson的问题，即强清洁环上的基体环是否再次强清洁。他们都证明了2 x 2矩阵环M2＆parl0; Z 2＆parr0;。是不是很干净的，其中Z2是Z在素理想值（2）的局部化。关于该问题，获得了以下结果：（1）强清洁环上非高度清洁的矩阵环的各种示例。 （2）完全确定M2（R）非常干净的局部环R（可交换或不可交换）。 （3）任意环R上的M2（R）强烈清洁的必要条件。 （4）当R具有IBN且每个有限生成的射影R-模是自由的时，Mn（R）中的单个矩阵强烈清洁的准则。 （5）足以使交换环R上的基环Mn（R）强烈清洁的条件。 （6）交换局部环R上的Mn（R）强烈清洁的必要和充分条件。 （7）一族高度清洁的三角形矩阵环。 （8）在非交换局部环或强pi正则环上的新的强pi正则（当然是强清洁）矩阵环族。
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