When $(D[[X]])_{P[[X]]}$ is a valuation ring
HTML articles powered by AMS MathViewer
- by J. T. Arnold and J. W. Brewer PDF
- Proc. Amer. Math. Soc. 37 (1973), 326-332 Request permission
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
- Paul M. Eakin Jr. and William J. Heinzer, Some open questions on minimal primes of a Krull domain, Canadian J. Math. 20 (1968), 1261–1264. MR 241398, DOI 10.4153/CJM-1968-122-6
- Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
- Robert Gilmer and William Heinzer, Rings of formal power series over a Krull domain, Math. Z. 106 (1968), 379–387. MR 232759, DOI 10.1007/BF01115087
Additional Information
- © Copyright 1973 American Mathematical Society
- 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