Let $X$ be a topological space and consider the ring $C(X,\mathbb{R})$ of continuous real-valued functions on $X$ (where ring-addition and multiplication are defined in the obvious point-wise way).


It is well-known that this ring uniquely determines the topological space, if the latter is compact and Hausdorff (Gelfand-Duality).


My question is the following: What properties does the ring $C(X,\mathbb{R})$ have, if $X$ is a Stone space (that is, if $X \in Comp\_2$ is also totally disconnected)? Ideally, I would like to have a statement along the following line:


$X$ is a Stone space (if and) only if the ring $C(X,\mathbb{R})$ has the property [nice property given to me by the lovely folks from MO].



The ring of continuous real-valued functions on a Stone space
At first Let me recall the notion of a clean ring:


**Definition**:A commutative ring $R$ is called clean if every element of $R$ is a sum of a unit and an idempotent.


The following theorem is due to F.Azarpanah who first studied The notion of cleanness in rings of continuous real valued functions. you can find the details of it in [This Article](http://www.springerlink.com/content/hcm6pkgg7k8j1vtp/) 


**Theorem1**:The following statements are equivalent:


* $C(X)$ is a **clean ring**.
* $C^\*(X)$ is a **clean ring**.
* $X$ is strongly **zero-dimensional**.(i.e. $\beta X$ is zero-dimensional or stone space. )
* Every zero-divisor in $C(X)$ is clean.
* $C(X)$ has a clean prime ideal.


I think the illustrious part the above theorem is the relation between cleanness of $C(X)$ and Zero-dimensionality of $\beta X$.


Now Lets turn to your Question. I think the following theorem relates stone spaces to the cleanness property of $C(X)$.


**Theorem2**:Let $X$ be a **compact space**. then $X$ is a Stone Space if and only if the ring $C(X)$ is a clean ring.


The proof is clear. because in this case $C(X)=C^\*(X)$ is clean iff $\beta X=X$ is Zero dimensional or iff $X$ is a Stone Space. 


