Extendibility criterion for a projective module of rank one over R[T] and R[T,T_-1]

Abstract

In this note we give a criterion for a finitely generated projective module P of constant rank one over R[ T ] or R[ T, T ] to be extended from R in terms of invertible ideals, when R is an integral domain. We show that if I is an invertible ideal of R[ T ] or R[ T, T ] such that I ∩ R ≠ 0, then I is extended from R if and only if I ∩ R is an invertible ideal of R.