Ratliff-Rush ideal and reduction numbers

TL;DR

This paper investigates the relationship between the Ratliff-Rush reduction number and the usual reduction number of an ideal in a Cohen-Macaulay local ring, providing an inequality that answers a previously posed question.

Contribution

It establishes that the Ratliff-Rush reduction number is less than or equal to the reduction number, addressing a question by Rossi and Swanson.

Findings

Proves $ ilde{r}_J(I) \\leq r_J(I)$ for m-primary ideals in Cohen-Macaulay rings.

Answers an open question by Rossi and Swanson.

Enhances understanding of reduction numbers in commutative algebra.

Abstract

Let $(R,m)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an m-primary ideal and $J$ a minimal reduction of $I$. In this paper, we show that $\widetilde{r_J(I)}\leq r_J(I)$. This answer to a question that made by M.E. Rossi and I. Swanson in [\ref{Rs}, Question 4.6].