Commutative ring satisfying a.c.c. and d.c.c. on radical ideals

Question

Let $R$ be a commutative ring with unity whose prime spectrum is both Noetherian and Artinian under Zariski topology i.e. $R$ satisfies a.c.c. and d.c.c. on radical ideals. Then is it true that $R$ has dimension zero ? If this is not true in general, is it true if we further assume $R$ is local ?

Answer 1

No. For instance, any finite topological space is trivially both Noetherian and Artinian, but need not be zero-dimensional. Explicitly, for instance, if $R$ is a discrete valuation ring then its spectrum has two points but is not zero-dimensional.