I asked this question at MSE but I did not received any answer, so I repeat it here at MO:
What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$ of all real (or complex) valued continuous functions on $X$ is a clean ring?
A clean ring is a ring in which every element is a sum of a unit and an idempotent.
Well, the two point set will do. Also an infinite discrete set will do. To give a less obvious example, let me argue that every compact space with a basis of clopen sets will do too.
Let $X$ be such and consider a real valued function $f$ on $X$. Consider the compact subset $\{f \leq 1/3\}$ and cover it with clopen sets that are contained in $\{f<2/3\}$. Choose a finite subcover and let $p$ be the characteristic function of its union. This is a continuous idempotent. $u=f-p$ is a continuous function that is bounded away from 0, hence a unit. $f=p+u$.
For a Tychonoff space $X$, the rings $C(X)$ and $C(X,\mathbb{C})$ are each clean if and only if the space $X$ is strongly zero-dimensional (s.z.d). This is a result of Azarpanah, me, and most recently Arora and Kundu.
A Tychonoff space $X$ is s.z.d if and only if every cozeroset is the countable union of clopen subsets. In particular, s.z.d implies a base of clopen sets. Examples of s.z.d spaces include the rationals, the Cantor space, and W.
