Let K be a field and R = K[x1,...,xn] be the polynomial ring in the variables x1, ..., xn. In this paper we prove that when a = {p1, ...,pm} and b={p'1,...p't} 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, AssR(R/J)=b, and AssR(J/I) =a\b, where Ass(I) is the stable set of associated prime ideals of I. Also we show that when p1,...,pm are nonzero monomial prime ideals of R generated by disjoint nonempty subsets of {x1,..., xn}, then there exists a square-free monomial ideal I such that AssR(R/I^k) = Ass(I) ={p1,...,pm} for all k ≥ 1.

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