A topological space $X$ is called profinite if it is compact, Hausdorff, and has a basis of open–closed sets. Also a commutative ring $R$ with 1 is called a topological ring it there is a topology on $R$ such that the following conditions hold:

1) $(z,y) \rightarrow z + y$ is continuous from $R\times R$ to $R$;

2) $z \rightarrow -z$ is continuous from $R$ to $R$;

3) $(z,y) \rightarrow zy$ is continuous from $R\times R$ to $R$.

Now let $C(X,R)$ be the ring of all continuous functions from $X$ to $R$, where $X$ is profinit and $R$ is a topological ring. I am looking for a reference for studying such rings specially ideal theory of such rings.

zero-dimensional(rarely,Boolean) [profinitesuggests an inverse limit of finite spaces, usually used with topological groups]. There is an AMS lecture notes by Pierce dealing with the rings and spaces. $\endgroup$ – David Handelman Dec 1 '18 at 22:40