Root extension in polynomial and power series rings

Abstract

An extension $R\subseteq S$ of commutative rings with unity is called a root extension if for each element $s\in S$, there exists a positive integer $n$ such that $s^n\in R$. Unlike the integral extension, the root extension is not stable under polynomial ring extension. We characterize when the extension $R\left[X\right]\subseteq S\left[X\right]$ of polynomial rings is a root extension. Using the characterization, we can give a positive answer to the question posed by Anderson, Dumitrescu and Zafrullah (2004), i.e., $R\left[X\right]\subseteq S\left[X\right]$ being a root extension implies that $R\left[X,Y\right]\subseteq S\left[X,Y\right]$ is a root extension. We also characterize when the extension $R\left[\left[X\right]\right]\subseteq S\left[\left[X\right]\right]$ of power series rings is a root extension.