On the Stable Set of Associated Prime Ideals of Monomial Ideals and Square-Free Monomial Ideals

Abstract

Let K be a field and R = K[x ₁,…, x _n ] be the polynomial ring in the variables x ₁,…, x _n . In this paper we prove that when 𝔄 = {𝔭₁,…, 𝔭 _m } and are two arbitrary sets of monomial prime ideals of R, then there exist monomial ideals I and J of R such that I ⊆ J, Ass^∞(I) = 𝔄 ∪ 𝔅, Ass _R (R/J) = 𝔅, and Ass _R (J/I) = 𝔄 \ 𝔅, where Ass^∞(I) is the stable set of associated prime ideals of I. Also we show that when 𝔭₁,…, 𝔭 _m are nonzero monomial prime ideals of R generated by disjoint nonempty subsets of {x ₁,…, x _n }, then there exists a square-free monomial ideal I such that Ass _R (R/I ^k ) = Ass^∞(I) = {𝔭₁,…, 𝔭 _m } for all k ≥ 1.

Key Words:

• Associated prime ideal
• Monomial ideal

2010 Mathematics Subject Classification:

• 13C13
• 13P99

ACKNOWLEDGMENTS

The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions.

Notes

Communicated by J. Alev.