Algebraic analysis of some strongly clean rings and their generalizations

Let R be an associative ring with identity and U( R) denote the set of units of R. An element a ∈ R is called clean if a = e + u for some e 2 = e and u ∈ U(R) 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). Let Z(R) be the center of R and g(x) be a polynomial in the polynomial ring Z(R)[x]. An element a ∈ R is called g( x)-clean if a = s + u where g(s) = 0 and u ∈ U(R) and a is called strongly g(x)-clean if, in addition, su = us. The ring R is called g( x)-clean (resp., strongly g(x)-clean) if every element of R is g(x)-clean (resp., strongly g(x)-clean). A ring R has stable range one if Ra + Rb = R with a, b ∈ R implies that a + yb ∈ U(R) for some y ∈ R.;In the process of settling these questions, we actually get: (1) The ring of continuous functions C(X) on a completely regular Hausdorff space X is strongly clean iff it has stable range one; (2) A unital C*-algebra with every unit element self-adjoint is clean iff it has stable range one; (3) Necessary conditions for the matrix rings Mn (R) (n ≥ 2) over an arbitrary ring R to be strongly clean; (4) Strongly clean property of M2 (RC2) with certain local ring R and cyclic group C2 = {1, g}; (5) A sufficient but not necessary condition for the matrix ring over a commutative ring to be strongly clean; (6) Strongly clean matrices over commutative projective-free rings or commutative rings having ULP; (7) A sufficient condition for Mn (C(X)) ( Mn (C(X, C ))) to be strongly clean; (8) If R is a ring and g(x) ∈ (x--a)( x--b)Z(R)[x] with a, b ∈ Z(R), then R is (x--a)(x--b)-clean iff R is clean and b -- a ∈ U(R), and consequently, R is g(x)-clean when R is clean and b -- a ∈ U( R); (9) If R is a ring and g( x) ∈ (x -- a)(x -- b)Z( R)[x] with a, b ∈ Z( R), then R is strongly (x -- a)( x -- b)-clean iff R is strongly clean and b -- a ∈ U(R), and consequently, R is strongly g(x)-clean when R is strongly clean and b -- a ∈ U( R).;In this thesis, we consider the following three questions: (1) Does every strongly clean ring have stable range one? (2) When is the matrix ring over a strongly clean ring strongly clean? (3) What are the relations between clean (resp., strongly clean) rings and g(x)-clean (resp., strongly g(x)-clean) rings?...