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

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 ?

• @user26857: Thanks for making my case in your comment ...
– user640299
Feb 20, 2019 at 21:08
• My apologies, close vote retracted. Feb 20, 2019 at 21:10

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.