# Integral Closure of Monomial Ideals on Regular Sequences

### Karlheinz Kiyek

Universität Paderborn, Germany### Jürgen Stückrad

Universität Leipzig, Germany

## Abstract

It is well known that the integral closure of a monomial ideal in a polynomial ring in a finite number of indeterminates over a field is a monomial ideal, again. Let $R$ be a noetherian ring, and let $(x_{1},…,x_{d})$ be a regular sequence in $R$ which is contained in the Jacobson radical of $R$. An ideal $a$ of $R$ is called a monomial ideal with respect to $(x_{1},…,x_{d})$ if it can be generated by monomials $x_{1}⋯x_{d}$. If $x_{1}R+⋯+x_{d}R$ is a radical ideal of $R$, then we show that the integral closure of a monomial ideal of $R$ is monomial, again. This result holds, in particular, for a regular local ring if $(x_{1},…,x_{d})$ is a regular system of parameters of $R$.

## Cite this article

Karlheinz Kiyek, Jürgen Stückrad, Integral Closure of Monomial Ideals on Regular Sequences. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 483–508

DOI 10.4171/RMI/359