Valuation overrings of a Noetherian domain

摘要

Let R be a Noetherian domain and 0 = P_0 ? P_1 ? ...? P_n be a saturated chain of prime ideals of R. Let V be a valuation overring of R that has a chain of prime ideals {Q_α}_(α∈A) such that{ Q_α∩ R}_(α∈A)= {P_i}_i~n=0. In this paper, we prove that {Q_α}_(α∈A = {0 = Q0 ? Q_1 ? ...? Q_n} and V_(Qn) is discrete， i.e.， Q_i V_(Qi)i is principal for all i = 1，...，n. Let D be an integral domain such that D_p is Noetherian for each prime ideal P of D with htP < ∞. As a corollary, we show that if {P_k} is a chain of prime ideals of D such that htP_k < ∞ for each k, then there exists a discrete valuation overring of D which has a chain of prime ideals lying over {P_k}.