Almost clean elements in C(X)

Abstract

An element a in a ring R is called almost clean if a can be written as a = r + e, where r is a regular (non zero-divisor) element and e is an idempotent. It is well known that C(X) is clean if and only if it is almost clean. In this paper a topological characterization of almost clean elements of C(X) is given and using this by a direct and short proof, it is shown that C(X) is clean if and only if it is almost clean. The coincidence of the various kind of cleanness of C(X) is studied. Whenever X is locally compact, we will show that CK(X) is almost clean if and only if C_K(X) is clean if and only if C_∞(X) is clean if and only if C_∞(X) is almost clean. It turns out that a prime ideal of C(X) is clean if and only if it is almost clean. We also show that   (which in the special case is equal to the super socle of C(X)) is clean and if X is a Lindelof weak P -space, then M^βX\I(X) is almost clean. We prove that X is a P -space if and only if C(X) is a von Neumann u-regular ring (we say that R is a von Neumann u-regular ring if a is a von Neumann regular element implies that 1 + a is a von Neumann regular element, for any a ∈ R). We observe that a ring R is a von Neumann regular ring if and only if it is clean and von Neumann u-regular. Finally, it is shown that if X is a connected space, then X is an almost P -space if and only if every almost clean element of C(X) is clean.