Algebraic analysis of some strongly clean rings and their generalizations

Abstract

Let R be an associative ring with identity and U(R) denote the set of unites of R. An element aεR is called clean if a = e + u for some e² = e and uε 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 Rand g(x) be a polynomial in the polynomial ring Z(R)[x]. An element aεR is called g(x)-clean if a = s + u where g(s) = 0 and uε 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, bεR implies that a+yb ε U(R) for some yεR. -- In this thesis, we consider the following three questions: -- Does every strongly clean ring have stable range one? -- When is the matrix ring over a strongly clean ring strongly clean? -- What are the relations between clean (resp., strongly clean) rings and g(x)-clean (resp., strongly g(x)-clean rings? -- In the process of settling these questions, we actually get: The ring of continuous functions C(X) on a completely regular Huasdorff space X is strongly clean if it has stable range one; -- A unital C*-algebra with every unit element self-adjoint is clean if it has stable range one; Necessary conditions for the matrix rings M₂(RC₂) (n ≥ 2) over an arbitrary ring R to be strongly clean; -- Strongly clean property of M₂(RC₂) with certain local ring R and cyclic group C₂ = {1,g}; A sufficient but not necessary condition for the matrix ring over a commutative ring to be strongly clean; -- Strongly clean matrices over commutative projective-free rings or commutative rings having ULP; -- A sufficient condition for Mn(C(X)) (Mn(C(X,C))) to be strongly clean; -- If R is a ring and g(x)ε(x-a)(x-b)Z(R)[x] with a, bεZ(R), then R is (x-a)(x-b) clean if R is clean and b-aεU(R), and consequently, R is g(x)-clean when R is clean and b-aεU(R); -- If R is a ring and g(x)ε(x-a)Z(R)[x] with a,bεZ(R), then R is strongly (x-a(x-b)-clean when R is strongly clean and b-aεU(R).