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On Krull’s Conjecture concerning Valuation Rings

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Mathematical Institute, Nagoya University
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Previously W. Krull conjectured that every completely integrally closed primary domain of integrity is a valuation ring, The main purpose of the present paper is to construct in §1 a counter example against this conjecture. In § 2 we show a necessary and sufficient condition that a field is a quotient field of a suitable completely integrally closed primary domain of integrity which is not a valuation ring.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 4 , June 1952 , pp. 29 - 33
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1952

References

1) Krull, W., Beitráge zur Arithmetik kommutativer Integritâtsbereiche II, Math, Zeit. 41 (1936). p, 670.Google Scholar

2) A ring is called primary if it has at most one proper prime ideal.

3) Observe the fact that 2α ∉ G, because K is algebraically closed.

4) Since 2x ∉ G, we is uniquely determined by the relation we(x + e) =2x.

5) If α = λ0 or 2α = λ0, we see easily that n — i0 = m— j 0 because α ∉ G. In this case, s = 0 is also clear.

6) Because o is integral over r, xo = x ∩ p = x ∩ q.

7) Krull, W., Beiträge zur Arithmetik kommutativer Integritätsbereiche, Math. Zeit. 41 (1936), Theorem 7 (p. 554).Google Scholar

8) Prüfer, Untersuchungen über die Teilbarkeitseigenschaften in Körpern, Crelle 168, p, 31 or 1. c. note 6) Theorem 8 (p. 555).