The plus closure of an ideal. (Q1429900)

Let \(R\) be a regular local ring with quotient field \(K\), \(I\) the ideal of \(R\) generated by a part \(x,y\) of a regular system of parameters and \(R^+\) the integral closure of \(R\) in an algebraic closure of \(K\). Assume that the characteristic \(p\) of the residue field of \(R\) is a sufficiently large prime number and \(p\in I\). Let \(z\in R^+\) with \(z^3\in R\). In the paper under review, the author shows that \(z\in IR^+\) if one of the following holds: (1) \(z^3\in t^3R\) for some \(t\in I\), (2) \(z^3\in I^6\), or (3) \(z^3\in (t,I^3)^4\cup (t,I^2)^5\cup (t^5,I^8)\) for some \(t\in I\). He also proves many cases of the converse. Now let \(R\) be an integrally closed domain, \(I\) a finitely generated ideal of \(R\) and \(z\in R^+\) with \(z^a\in R\) for some \(a\geq 1\). Sufficient conditions for \(z\in IR^+\) are also given.