Integral Closure of Monomial Ideals on Regular Sequences

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 $(x_1,\dots ,x_d)$ 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 $(x_1,\dots ,x_d)$ if it can be generated by monomials $x_1^{i_1}\cdots x_d^{i_d}$. If $x_{1}R+\cdots +x_{d}R$ 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 $(x_1,\dots ,x_d)$ is a regular system of parameters of $R$.