Strong Artin-Rees Property in Rings of Dimension One and Two

Abstract

Let (R, m) be a local noetherian ring and let N \subseteq M be two finitely generated R-modules such that the dim M/N \leq 1. We give simple proof of the fact that there exists an integer h such that I^n M \cap N = I^{n-h}(I^hM \cap N), for all n \geq h and for all ideals I \subset R. We give upper bounds for such an integer h. Moreover, we give two examples of rings of dimension two where the property fails.