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 ?
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1$\begingroup$ @user26857: Thanks for making my case in your comment ... $\endgroup$– user640299Feb 20, 2019 at 21:08
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$\begingroup$ My apologies, close vote retracted. $\endgroup$– KReiserFeb 20, 2019 at 21:10