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.

  • $\begingroup$ 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. $\endgroup$ – David Handelman Dec 1 '18 at 22:40

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.