## Revista Matemática Iberoamericana

### Asymptotic behaviour of monomial ideals on regular sequences

Monireh Sedghi

#### Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathbf{x}= x_1, \ldots, x_d$ 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 $x_1^{e_1}\ldots x_d^{e_d}$. The monomial closure of $I$, denoted by $\widetilde{I}$, 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 $\mathrm{Ass}_RR/\widetilde{I^n}$ and $\mathrm{Ass}_R\widetilde{I^n}/I^n$, $n=1,2, \ldots,$ 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.

#### Article information

Source
Rev. Mat. Iberoamericana, Volume 22, Number 3 (2006), 955-962.

Dates
First available in Project Euclid: 22 January 2007

https://projecteuclid.org/euclid.rmi/1169480036

Mathematical Reviews number (MathSciNet)
MR2320407

Zentralblatt MATH identifier
1115.13016

Subjects
Primary: 13B20 13B21: Integral dependence; going up, going down

#### Citation

Sedghi, Monireh. Asymptotic behaviour of monomial ideals on regular sequences. Rev. Mat. Iberoamericana 22 (2006), no. 3, 955--962. https://projecteuclid.org/euclid.rmi/1169480036

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