The completion of a ring with a valuation

Adams, Helen E.

doi:10.1090/S0002-9939-1972-0357519-4

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The completion of a ring with a valuation
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by Helen E. Adams PDF

Proc. Amer. Math. Soc. 36 (1972), 21-25 Request permission

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

Helen E. Adams, Filtrations and valuations on rings, Pacific J. Math. 43 (1972), 7–14. MR 318240
D. K. Harrison, Finite and infinite primes for rings and fields, Mem. Amer. Math. Soc. 68 (1966), 62. MR 207735
Merle E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193–198. MR 233813, DOI 10.1090/S0002-9939-1969-0233813-2