Note on Krull's conjecture

Krull in [7] conjectured that the answer to this conjecture is true, at least for the case where F is the quotient field of D and D is completely integrally closed. Nakayama [9, 10], Ohm (cf. [5, p. 232]) and Sheldon [12] gave counter examples to the conjecture. Krull proved that the conjecture holds true for one dimensional completely integrally closed quasi-local domains [8, Satz 1]. In this paper, among other things, we will prove the following facts: we characterize one dimensional Prufer domains (Corollary 2). Based on Gilmer's result [6], we prove that if F is an extension field of the quotient field K of D, then C(D), the complete integral closure of D, is the intersection of valuation