Asymptotic Primes of Ratliff–Rush Closure of Ideals with Respect to Modules

Abstract

Let R be a commutative Noetherian ring, M a nonzero finitely generated R-module, and I an ideal of R. The purpose of this article is to develop the concept of Ratliff–Rush closure of I with respect to M. It is shown that the sequence , n = 1,2,…, of associated prime ideals is increasing and eventually stabilizes. This result extends Mirbagheri–Ratliff's main result in Mirbagheri and Ratliff (1987). Furthermore, if R is local, then the operation is a c*-operation on the set of ideals I of R, each ideal I has a minimal Ratliff–Rush reduction J with respect to M, and, if K is an ideal between J and I, then every minimal generating set for J extends to a minimal generating set of K.

Key Words:

• Associated primes
• c*-Operation
• Ratliff–Rush closure
• Ratliff–Rush reduction
• Rees ring

2000 Mathematics Subject Classification:

• 13B20
• 13B21
• 13E05

ACKNOWLEDGMENTS

The authors are deeply grateful to the referee for his or her very careful reading of the original manuscript and valuable suggestions. The authors also would like to thank Professor Rahim Zaare-Nahandi for his careful reading of the first draft and many helpful suggestions.

This research of the second author was been in part supported by a grant from IPM (No. 84130042).

Notes

Communicated by I. Swanson.