Some Remarks About Saturation of Ideals

Abstract

In this paper we observe when saturation of ideals in a local ring R commutes with extension along ring maps and initial ideals. We give a characterization in terms of Cohen Macaulayness for when this happens along the map from R to R modulo its nilradical. We give several examples and non examples where this happens, and we demonstrate an application of this condition to epsilon multiplicity. Additionally, we show that saturation commutes with extension along a flat injection of local rings if and only if the closed fiber of the injection is Artinian.