# Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals

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.

• 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? – Christopher Oct 3 '18 at 15:35