ROOT EXTENSION IN POLYNOMIAL AND POWER SERIES RINGS

초록

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

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