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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

When $ (D[[X]])\sb{P[[X]]}$ is a valuation ring


Authors: J. T. Arnold and J. W. Brewer
Journal: Proc. Amer. Math. Soc. 37 (1973), 326-332
MSC: Primary 13F20; Secondary 13A15
DOI: https://doi.org/10.1090/S0002-9939-1973-0311656-X
MathSciNet review: 0311656
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Abstract: Let D be an integral domain with identity and let K denote the quotient field of D. If P is a prime ideal of D denote by $ P[[X]]$ that prime ideal of $ D[[X]]$ consisting of all those formal power series each of whose coefficients belongs to P. In this paper the following question is considered: When is $ {(D[[X]])_{P[[X]]}}$ a valuation ring? Our main theorem states that if $ {(D[[X]])_{P[[X]]}}$ is a valuation ring, then $ {D_P}$ must be a rank one discrete valuation ring. Moreover, we show that if $ {D_P}$ is a rank one discrete valuation ring and if $ PD[[X]] = P[[X]]$, then $ {(D[[X]])_{P[[X]]}}$ is a valuation ring. We also give an example to show that $ {(D[[X]])_{P[[X]]}}$ need not be a valuation ring when $ {D_P}$ is rank one discrete.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0311656-X
Keywords: Formal power series, valuation ring
Article copyright: © Copyright 1973 American Mathematical Society