Topologically boolean and g(x)-clean rings

Abstract

Let R be a ring with identity and let g(x) be a polynomial in Z(R)[x] where Z(R) denotes the center of R. An element r ? R is called g(x)-clean if r = u + s for some u,s ? R such that u is a unit and g(s) = 0. The ring R is g(x)-clean if every element of R is g(x)-clean. We consider g(x) = x(x?c) where c is a unit in R such that every root of g(x) is central in R. We show, via set-theoretic topology, that among conditions equivalent to R being g(x)-clean, is that R is right (left) c-topologically boolean.