Integral closure of monomial ideals on regular sequences (Q1884049)

Let \(K\) be a field and \(X_1\),...,\(X_n\) indeterminates. It is well known that the integral closure of a monomial ideal of \(K[X_1,\dots,X_n]\) is again a monomial ideal. Let \(R\) be a Noetherian ring and \(x_1,\dots,x_d\) a regular sequence in \(R\) contained in the Jacobson radical of \(R\). An ideal of \(R\) is said to be monomial with respect to \(x_1,\dots,x_d\) if it can be generated by monomials \(x_1^{i_1}\cdots x_d^{i_d}\). Under the assumption that \(x_1R+\cdots +x_dR\) is a radical ideal, the authors prove that the integral closure of a monomial ideal is still a monomial ideal; more precisely, if \(I\) is a monomial ideal, then its integral closure is generated by all monomials \(m\) with \(m^k\in I^k\) for some \(k\).