Cancellation ideals of a ring extension

We study properties of cancellation ideals of ring extensions. Let $R \subseteq S$ be a ring extension. A nonzero $S$-regular ideal $I$ of $R$ is called a (quasi)-cancellation ideal of the ring extension $R \subseteq S$ 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 $R\subseteq S$ if and only if $I$ is $S$-invertible.