A test theorem on coherent GCD domains

Abstract

Let R R be a commutative indecomposable coherent ring. Then the following statements are equivalent: (i) R R is a GCD domain; (ii) R M {R_M} is a GCD domain for every maximal ideal of M M of R R , and every finitely generated projective ideal in R R is principal; (iii) every two-generated ideal in R R has finite projective dimension, and every finitely generated projective ideal in R R is principal. Auslander-Buchsbaum’s Theorem, etc. can be obtained from the result above.