Lemma 10.119.12 (Krull-Akizuki). Let R be a domain with fraction field K. Let L/K be a finite extension of fields. Assume R is Noetherian and \dim (R) = 1. In this case any ring A with R \subset A \subset L is Noetherian.
Proof. To begin we may assume that L is the fraction field of A by replacing L by the fraction field of A if necessary. Let I \subset A be a nonzero ideal. Clearly I generates L as a K-vector space. Hence we see that I \cap R \not= (0). Pick any nonzero x \in I \cap R. Then we get I/xA \subset A/xA. By Lemma 10.119.11 the R-module A/xA has finite length as an R-module. Hence I/xA has finite length as an R-module. Hence I is finitely generated as an ideal in A. \square
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