Subintegrality, invertible modules and polynomial extensions

Motivated by a result of Traverso and Swan on seminormality, we prove that for a ring extension A⊆BA⊆B, A is subintegrally closed in B if and only if the group of invertible A-submodules of B   is canonically isomorphic to the group of invertible A[X]A[X]-submodules of B[X]B[X]. We also examine a relationship between these two groups in the general case, i.e. when A may not be subintegrally closed in B.