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 ?={?(1),..., ?(m)} and <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="lagb_a_793696_o_ilm0001.gif"></inline-graphic> are two arbitrary sets of monomial prime ideals of R, then there exist monomial ideals I and J of R such that IJ, Ass(I)=??, Ass(R)(R/J)=?, and Ass(R)(J/I)=?\?, where Ass(I) is the stable set of associated prime ideals of I. Also we show that when ?(1),..., ?(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)={?(1),..., ?(m)} for all k1.