Abstract
An extension R⊆S of commutative rings with unity is called a root extension if for each element s∈S, there exists a positive integer n such that sn∈R. Unlike the integral extension, the root extension is not stable under polynomial ring extension. We characterize when the extension R[X]⊆S[X] 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[X]⊆S[X] being a root extension implies that R[X,Y]⊆S[X,Y] is a root extension. We also characterize when the extension R[[X]]⊆S[[X]] of power series rings is a root extension.
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