A test theorem on coherent GCD domains

Abstract

Abstract.Let R be a commutative indecomposable coherent ring. Then the following statements are equivalent: (i) R is a GCD domain; (ii) RM 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.Let Ä bea commutative ring with 1 ^ 0 in this paper. R is called a GCD domain if every two elements a, b € R have a greatest common divisor (denoted by [a, b]) in R. R is said to be indecomposable if 1 is the only nonzero idempotent of R. A coherent ring is a ring whose finitely generated ideals are finitely presented.Definition. We say that R has PPC if every finitely generated projective ideal in R is a principal ideal.Theorem. Let R be an indecomposable coherent ring. Then the following statements are equivalent:(i) R is a GCD domain.(ii) Rm is a GCD domain for every maximal ideal M of R, and R has PPC.(iii) Every two-generated ideal in R has finite projective dimension, and R has PPC. Proof of Theorem, (i) =s> (ii) Let M be a maximal ideal of R. Let f , * e R, and write c = [a, b]. if y is a common divisor of y and * in Rm then d\sa