Considering semi-clean rings of continuous functions

Abstract

In this article we investigate commutative unital rings with emphasis on the ring of continuous complex-valued functions on a Tychonoff space X, denoted C(X,C), and its subring of real-valued continuous functions C(X). Clean rings are those in which every element is the sum of a unit and an idempotent. Semi-clean rings are those in which elements are sums of invertible elements and periodic elements. It is known that C(X,C) is clean if and only if C(X) is clean. The semi-cleanliness of C(X,C) is characterized in terms of the existence of the separation of continuous functions to [0,1]. We investigate when the embedding of C(X) into C(X,C) is an (associate) p-extension; work by Canfell is pivotal [7]. We exhibit an example of a p-extension which is not an associate p-extension; to our knowledge this is new. We demonstrate that spaces for which C(X,C) are semi-clean are necessarily totally disconnected. We conjecture that such spaces need not be strongly zero dimensional.