...Theorem 1. Let A be a principal ideal ring, and L a finite separable extension of its quotient field, of degree n. Let B be the integral closure of A in L. Then B is a free module of rank n over A. Proof. As a module over A, the integral closure is torsion-free,...

...finitely generated as a K -algebra, and such that L is separable over the field of fractions of A. Let B be the integral closure of A in L. Then B is finitely generated as a K -algebra, hence Noetherian. Proof. By the Noether Normalization Lemma...