Recall that a commutative ring $R$ with identity is a chain ring if the set of ideals of $R$ is linearly ordered under inclusion.
I want to know if there a chain ring with Krull dimension greater than one. Or does every chain ring have Krull dimension at most one?
The short answer is that you can get chained rings (or even valuation domains) of arbitrarily high Krull dimension. Without going into full details, examples can be created with constructions involving monoid rings. One way that valuation domains of Krull dimension one are special is that they are precisely the valuation domains that are completely integrally closed.