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 $\mathcal {P}$ of constant rank one over $R[T]$ or $R[T,{T^{ - 1}}]$ 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 $R[T,{T^{ - 1}}]$ such that $I \cap R \ne 0$, then $I$ is extended from $R$ if and only if $I \cap R$ is an invertible ideal of $R$.