Asymptotic behaviour of monomial ideals on regular sequences

Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathbf{x}=x_1,\dots ,x_d$ 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 $\mathbf{x}$ if it is generated by a set of monomials $x_1^{e_1}\dots x_d^{e_d}$. The monomial closure of $I$, denoted by $\tilde{I}$, is defined to be the ideal generated by the set of all monomials $m$ such that $m^n\in I^n$ for some $n\in \mathbb{N}$. It is shown that the sequences ${\mathrm{Ass}}_{R}R/\widetilde{I^n}$ and ${\mathrm{Ass}}_R\widetilde{I^n}/I^n$, $n=1,2,\dots ,$ 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.