We denote The rings of all continuous Real valued functions on $X$ by $C(X)$. also all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of $C(X)$ is to connect algebraic properties of the Ring $C(X)$ with the topological properties of the space $X$.

For a simple and well-Known connection we could see the following theorem:

Theorem: the topological space $X$ is connected iff the ring $C(X)$ has no idempotent element other than $0$ and $1$.

My Question comes from the relation between ring homomorphisms of the rings $C(X)$,$C(Y)$ and topological spaces $X,Y$.

$Q_1$:Let $X$ and $Y$ be two $T_{3\frac{1}{2}}$topological spaces. If $X$ is a zero dimensional space and $C(X)$ is ring isomorphic to $C(Y)$.(i.e.$C(X)\cong C(Y)$) Can we deduce that $Y$ is also Zero dimensional space.

we recall that a topological space $X$ is zero dimensional if it has a base of clopen subsets.

The same Question Can be Asked by Changing the ring $C(X)$ with the Banach algebra $C_b(X)$ as follows:

$Q_2$:Let $X$ and $Y$ be topological spaces with the previous assumption. If $X$ is a zero dimensional space and $C_b(X)$ is ring isomorphic to $C_b(Y)$.(i.e.$C_b(X)\cong C_b(Y)$) Can we deduce that Y is also Zero dimensional space.