A test theorem on coherent GCD domains

Abstract:

Let $R$ be a commutative indecomposable coherent ring. Then the following statements are equivalent: (i) $R$ is a GCD domain; (ii) ${R_M}$ is a GCD domain for every maximal ideal of $M$ of $R$, and every finitely generated projective ideal in $R$ is principal; (iii) 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.