Associated Primes of Powers of Monomial Ideals: A Survey

Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R=Ik) ⊆ AssR(R=Ik+1) for all positive integers k, where AssR(R=I) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (Ik+1 : RI) = Ik for all positive integers k. Also, an ideal I is called normally torsion-free if AssR(R=Ik) ⊆ AssR (R=I) for all positive integers k. In this paper, we collect the latest results in associated primes of powers of monomial ideals in three concepts, i.e., the persistence property, strong persistence property, and normally torsion-freeness. Also, we present some classes of monomial ideals such that are none of edge ideals, cover ideals, and polymatroidal ideals, but satisfy the persistence property and strong persistence property. In particular, we study the Alexander dual of path ideals of unrooted starlike trees. Furthermore, we probe the normally torsion-freeness of the Alexander dual of some path ideals which are related to banana trees. We close this paper with exploring the normally torsion-freeness under some monomial operations.