Ratliff–Rush ideal and reduction numbers

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 $\widetilde{r_J\left(I\right)}\le r_J\left(I\right)$. This answer to a question that made by Rossi and Swanson in [15, Question 4.6].

KEYWORDS:

• Ratliff–Rush ideal
• reduction number

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C14
• 13H10
• 13D40

Acknowledgments

This paper was done while I was visiting the University of Osnabruck. I would like to thank the Institute of Mathematics of the University of Osnabruck for hospitality. Also, I would like to express my deep gratitude to Professor Louis Ratliff and Professor Tony Puthenpurakal for valuable suggestions. Finally, I would like grateful to the referee for the careful reading of the manuscript and the helpful suggestions.