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The completion of a ring with a valuation


Author: Helen E. Adams
Journal: Proc. Amer. Math. Soc. 36 (1972), 21-25
MSC: Primary 16A80
DOI: https://doi.org/10.1090/S0002-9939-1972-0357519-4
MathSciNet review: 0357519
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Abstract: This paper proves three main results: the completion of a commutative ring with respect to a Manis valuation is an integral domain; a necessary and sufficient condition is given that the completion be a field; and the completion is a field when the valuation is Harrison and the value group is archimedean ordered.


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

  • [1] Helen E. Adams, Filtrations and valuations on rings, Pacific J. Math. 41 (1972). MR 0318240 (47:6787)
  • [2] D. K. Harrison, Finite and infinite primes for rings and fields, Mem. Amer. Math. Soc. No. 68 (1966), 62 pp. MR 34 #7550. MR 0207735 (34:7550)
  • [3] Merle E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193-198. MR 38 #2134. MR 0233813 (38:2134)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0357519-4
Keywords: Valuation on a ring, valuated ring completion, Manis valuation, Harrison prime
Article copyright: © Copyright 1972 American Mathematical Society

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