Root and integral closure for \(R[[X]]\) (Q788055)

Let R be a subring of S. R is said to be n-root closed in S if whenever \(s^ n\in R\), then \(s\in R\). R is n-root closed in S if R is n-root closed for every integer \(n>1\). The author shows that \({\mathbb{Z}}[[X]]\) is not n-root closed in \({\mathbb{Q}}[[X]]\) for any n, even though \({\mathbb{Z}}\) is root closed in \({\mathbb{Q}}\). If R is von Neumann regular, then integral closure, n-root closure and root closure of R in S are preserved for R[[X]] in S[[X]]. However, these properties may not be preserved for R[[X]] in its total quotient ring. The author investigates several conditions on R so that closure will be preserved in the power series ring.