  In 1969 H\"ochster proved that for every quasi-compact T1-space $X$ we can
find a commutative ring $R$ such that $X$ is homeomorphic to the maximal
spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a
commutative ring $R$ that admits a non-Hausdorff and totally disconnected
maximal spectrum $\mathrm{Specm}(R)$. However, there has not been an example of
such a commutative ring yet. The aim of this paper is to investigate rings that
almost admit a non-Hausdorff and totally disconnected maximal spectrum.
