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 $R$.