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

LetKbe a field andR = K[x1,…,xn] be the polynomial ring in the variablesx1,…,xn. In this paper we prove that when 𝔄 = {𝔭1,…, 𝔭m} andare two arbitrary sets of monomial prime ideals ofR, then there exist monomial idealsIandJofRsuch thatI ⊆ 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 ofI. Also we show that when 𝔭1,…, 𝔭mare nonzero monomial prime ideals ofRgenerated by disjoint nonempty subsets of {x1,…,xn}, then there exists a square-free monomial idealIsuch that AssR(R/Ik) = Ass∞(I) = {𝔭1,…, 𝔭m} for allk ≥ 1. [ABSTRACT FROM PUBLISHER]