Derivations and root closures of graded ideals (Q6635919)

The authors study the three root closures introduced by J. Forsman, one of the authors of the present paper. Here are two of the results in this paper.\N\NAssume that \(R\) is a commutative Noetherian ring. and let \(I\) be an ideal of \(R\). Then:\N\N\begin{itemize}\N\item If the ring \(R=\bigoplus_{i=0}^{\infty}R_{i}\) is graded, all the residue fields of \(R_{0}\) are infinite, and the ideal \(I\) is graded, then the three root closures of the ideal \(I\) are graded.\N\N\item If \(R\) is an integral domain containing a field of characteristic zero, and \(\delta\) is a locally nilpotent derivation of \(R\) such that \(\delta(I)\subseteq I\), then all three root closures of \(I\) are invariant under \(\delta\).\N\end{itemize}