| Summary: | Thesis (Ph.D.)--Memorial University of Newfoundland, 2009. Mathematics and Statistics Includes bibliographical references (leaves 66-72) Let R be an associative ring with identity and U(R) denote the set of units of R. An element α∈R is called clean if a = e + u for some e2 = e and u ∈ U(R) and α 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 α∈R is called g(x)-clean if a = s + u where g(s) = 0 and u ∈ U(R) and α 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 α, b ∈R implies that α + yb ∈U(R) for some y∈R. -- 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 Hausdorff 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 Mn(R) (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) ∈ (x-a)(x-b)Z(R)[x] with a, b ∈ Z(R), then R is (x-a)(x-b)-clean if R is clean and b - a ∈U(R), and consequently, R is g(x)-clean when R is clean and b - a ∈U(R); -- • 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 if 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).
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