On Rings Whose Elements are the Sum of a Unit and a Root of a Fixed Polynomial |
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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: | A ring R with identity is called “clean” if for every element a ? R there exist an idempotent e and a unit u in R such that a = e + u. Let C(R) denote the center of a ring R and g(x) be a polynomial in the polynomial ring C(R)x]. An element r ? R is called “g(x)-clean” if r = s + u where g(s) = 0 and u is a unit of R and R is g(x)-clean if every element is g(x)-clean. Clean rings are g(x)-clean where g(x) ? (x ? a)(x ? b)C(R)x] with a, b ? C(R) and b ? a ? U(R); equivalent conditions for (x2 ? 2x)-clean rings are obtained; and some properties of g(x)-clean rings are given. |
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Keywords: | Clean ring g(x)-Clean ring Matrix ring |
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