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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Algebraic extensions of power series rings


Author: Jimmy T. Arnold
Journal: Trans. Amer. Math. Soc. 267 (1981), 95-110
MSC: Primary 13J05; Secondary 13G05
MathSciNet review: 621975
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0621975-4
PII: S 0002-9947(1981)0621975-4
Keywords: Power series ring, algebraic extension, integral extension
Article copyright: © Copyright 1981 American Mathematical Society