Cancellation ideals of a ring extension

Abstract

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.