Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 0311656
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] P. M. Eakin, Jr. and W. J. Heinzer, Some open questions on minimal primes of a Krull domain, Canad. J. Math. 20 (1968), 1261-1264. MR 39 #2738. MR 0241398 (39:2738)
  • [2] R. Gilmer, Multiplicative ideal theory, Queens' Papers in Pure and Appl. Math., no. 12, Queen's University, Kingston, Ont., 1968. MR 37 #5198. MR 0229624 (37:5198)
  • [3] R. Gilmer and W. Heinzer, Rings of formal power series over a Krull domain, Math. Z. 106 (1968), 379-387. MR 38 #1082. MR 0232759 (38:1082)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13F20, 13A15

Retrieve articles in all journals with MSC: 13F20, 13A15

Additional Information

Keywords: Formal power series, valuation ring
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society