Cancellation ideals of a ring extension

Abstract

We study properties of cancellation ideals of ring extensions. Let R ⊆ S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R ⊆ S if whenever IB = IC for two S-regular (finitely generated) R-submodules B and C of S, then B = C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R ⊆ S if and only if I is S-invertible.