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Let $R$ be a Valuation ring (https://en.wikipedia.org/wiki/Valuation_ring) .

1) If $R$ satisfies a.c.c. on prime ideals, then does $R$ have finite Krull dimension ?

2) If $R$ satisfies d.c.c. on prime ideals, then does $R$ have finite Krull dimension ?

I know that $R$ satisfies both a.c.c. and d.c.c. on prime ideals, then $R$ has only finitely many prime ideals due to the comparability of any two ideals in a Valuation ring. But I don't know what happens if we restrict our condition to either only a.c.c. or only d.c.c.

Please help.

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    $\begingroup$ If either a.c.c. or d.c.c. on prime ideals fails, then $R$ has infinite Krull dimension. So the question reduces to whether either a.c.c or d.c.c (on prime ideals) implies the other for valuation rings? $\endgroup$ – Christopher Oct 3 '18 at 15:35

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