Semiclean rings and rings of continuous functions

Abstract

As defined by Ye [{\bf12}], a ring is semiclean if every element is the sum of a unit and a periodic element. Ahn and Anderson [{\bf1}] called a ring {weakly clean} if every element can be written as $u+e$ or $u-e$, where $u$ is a unit and $e$ an idempotent. A weakly clean ring is {semiclean}. We show the existence of semiclean rings that are not weakly clean. Every semiclean ring is $2$-clean. New classes of semiclean subrings of $\r$ and $\c$ are introduced and conditions are given when these rings are clean. Cleanliness and related properties of $C(X,A)$ are studied when $A$ is a dense semiclean subring of $\r$ or $\c$.