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 is spectral if and only if where are finite and
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"!