Is there a complete characterization of such spaces?
Thanks!
yes, there are many of them. these topological spaces are called spectral spaces. a nice characterization is in terms of inverse limit of topological
spaces: a topological space $\displaystyle X$ is spectral if and only if $\displaystyle X= \varprojlim Y_n,$ where $\displaystyle Y_n$ are finite and $\displaystyle T_0.$
i think now that you have a name, you can find other characterizations of spectral spaces.
by the way, i've assumed that by "ring" you meant "commutative ring"!