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An integral domain D satisfies ACC on principal ideals (ACJCP) if there does not exist an infinite strictly ascending chain of principal ideals of D. Any Noetherian domain, in particular any Dedekind domain, satisfies ACCP. In this note we prove the following theorem: Let D be an integral domain. Then the integral closure of D is a Dedekind domain if and only if every overring of D (ring between D and its quotient field) satisfies ACCP.
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