A Test Theorem on Coherent GCD Domains

Abstract

Let R be a commutative indecomposable coherent ring. The following statements are equivalent : R is a GCD domain; R M % is a GCD domain for every maxima ideal of M of R, and every finitely generated projective ideal in R is principal; every two-generated ideal in R has finite projective dimension, and every finitely generated projective ideal in R is principal. Auslander-Buchsbaum's Theorem, etc. can be obtained from the result above