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

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.