Revista Matemática Iberoamericana


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Volume 19, Issue 2, 2003, pp. 483–508
DOI: 10.4171/RMI/359

Published online: 2003-08-31

Integral Closure of Monomial Ideals on Regular Sequences

Karlheinz Kiyek[1] and Jürgen Stückrad[2]

(1) Universität Paderborn, Germany
(2) Universität Leipzig, Germany

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 $(x_1,\ldots,x_d)$ be a regular sequence in $R$ which is contained in the Jacobson radical of $R$. An ideal $\mathfrak a$ of $R$ is called a monomial ideal with respect to $(x_1,\ldots,x_d)$ if it can be generated by monomials $x_1^{i_1}\cdots x_d^{i_d}$. If $x_1R+\cdots + x_dR$ 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 $(x_1,\ldots,x_d)$ is a regular system of parameters of $R$.

Keywords: Regular sequences, monomial ideals, integral closure of monomial ideals

Kiyek Karlheinz, Stückrad Jürgen: Integral Closure of Monomial Ideals on Regular Sequences. Rev. Mat. Iberoam. 19 (2003), 483-508. doi: 10.4171/RMI/359