Strongly Clean Property and Stable Range One of Some Rings |
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Authors: | Lingling Fan |
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Institution: | Department of Mathematics and Statistics , Memorial University of Newfoundland , St. John's , Canada |
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Abstract: | Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R, and a is called strongly clean if, in addition, eu = ue. A ring R is clean if every element of R is clean, and R is strongly clean if every element of R is strongly clean. When is a matrix ring over a strongly clean ring strongly clean? Does a strongly clean ring have stable range one? For these open questions, we prove that 𝕄 n (C(X)) is strongly π-regular (hence, strongly clean) where C(X) is the ring of all real valued continuous functions on X with X a P-space; C(X) is clean iff it has stable range one; and a unital C*-algebra in which every unit element is self-adjoint is clean iff it has stable range one. The criteria for the ring of complex valued continuous functions C(X,?) to be strongly clean is given. |
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Keywords: | C*-algebra Clean ring Matrix ring Ring of continuous functions C(X) Ring of continuous functions C(X ?) Stable range one Strongly clean ring |
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