Ring with spectrum homeomorphic to a given topological space I would like to ask whether given a topological space $X$, we can find a commutative ring with unity $R$ such that $\operatorname{Spec} R$ (together with the Zariski topology) is homeomorphic to $X$.
Since the spectrum is a compact space, this is obviously only possible if $X$ is compact. Furthermore, from this answer we obtain that for spectra, $T_1$ already implies Hausdorff.
How many more restrictions must we impose? Can we give a characterisation of when a topological space is a spectrum of a ring?

You can check this page : https://en.wikipedia.org/wiki/Spectral_space which provides a pretty satisfactory answer.
