# A reference for studying special ring

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.

• The standard name for a topological space with a basis of clopen sets is zero-dimensional (rarely, Boolean) [profinite suggests 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. – David Handelman Dec 1 '18 at 22:40