Asymptotic behaviour of monomial ideals on regular sequences

Let R be a commutative Noetherian ring, and let x = x1, . . . , xd be a regular R-sequence contained in the Jacobson radical of R. An ideal I of R is said to be a monomial ideal with respect to x if it is generated by a set of monomials x1 1 . . . x ed d . The monomial closure of I, denoted by Ĩ, is defined to be the ideal generated by the set of all monomials m such that mn ∈ In for some n ∈ N. It is shown that the sequences AssRR/Ĩ and AssRĨn/I, n = 1, 2, . . . , of associated prime ideals are increasing and ultimately constant for large n. In addition, some results about the monomial ideals and their integral closures are included.