Square-free monomial ideals and stability of certain set of associated prime ideals

Let R be a commutative Noetherian ring and I be an ideal of R. A well-known result of Brodmann see [1] showed that the sequence AssR(R/Ik)k1 of associated prime ideals is stationary for large k, i.e., there exists a positive integer k0 such that AssR(R/Ik) = AssR(R/Ik0) for all k k0. A minimal such k0 is called the index of stability of I and AssR(R/Ik0) is called the stable set of associated prime ideals of I, which is denoted by Ass(I). Also we say an ideal I of R satises the persistence property if AssR(R/Ik) AssR(R/Ik+1) for all positive integers k. In this talk, we rst focus on the stable set of square-free monomial ideals, and state some results in this subject, see [2, 3]. We next present two classes of monomial ideals such that are none of edge ideals, cover ideals and polymatroidal ideals, but satisfy the persistence property, see [5]. We nally extend the notion of the persistence property for monomial ideals to a family of ideals in commutative Noetherian rings, see [4].