# Asymptotic behaviour of monomial ideals on regular sequences

### Monireh Sedghi

Azarbaidjan University of Tarbiat Moallem, Tabriz, Iran

## Abstract

Let $R$ be a commutative Noetherian ring, and let $x=x_{1},…,x_{d}$ 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 $x$ if it is generated by a set of monomials $x_{1}…x_{d}$. The monomial closure of $I$, denoted by $I$, is defined to be the ideal generated by the set of all monomials $m$ such that $m_{n}∈I_{n}$ for some $n∈N$. It is shown that the sequences $Ass_{R}R/I_{n}$ and $Ass_{R}I_{n}/I_{n}$, $n=1,2,…,$ 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.

## Cite this article

Monireh Sedghi, Asymptotic behaviour of monomial ideals on regular sequences. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 955–962

DOI 10.4171/RMI/479