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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Algebraic extensions of power series rings
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by Jimmy T. Arnold PDF
Trans. Amer. Math. Soc. 267 (1981), 95-110 Request permission

Abstract:

Let $D$ and $J$ be integral domains such that $D \subset J$ and $J[[X]]$ is not algebraic over $D[[X]]$. Is it necessarily the case that there exists an integral domain $R$ such that $D[[X]] \subset R \subseteq J[[X]]$ and $R \cong D[[X]][[\{ {Y_i}\} _{i = 1}^\infty ]]$? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if $D$ satisfies any one of the conditions (1) $D$ is Noetherian, (2) $D$ is integrally closed, (3) the quotient field $K$ of $D$ is countably generated as a ring over $D$, or (4) $D$ has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that $J[[X]]$ is algebraic over $D[[X]]$ if and only if it is integral over $D[[X]]$ and necessary and sufficient conditions are given on $D$ and $J$ in order that this occur. Finally if, for every positive integer $n$, $D[[{X_1}, \ldots ,{X_n}]] \subset R \subseteq J[[{X_1}, \ldots ,{X_n}]]$ implies that $R \ncong D[[{X_1}, \ldots ,{X_n}]][[\{ {Y_i}\} _{i = 1}^\infty ]]$, then it is shown that $J[[{X_1}, \ldots ,{X_n}]]$ is algebraic over $D[[{X_1}, \ldots ,{X_n}]]$ for every $n$.
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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 267 (1981), 95-110
  • MSC: Primary 13J05; Secondary 13G05
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0621975-4
  • MathSciNet review: 621975