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On the Theory of Henselian Rings, II

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Mathematical Institute, Kyoto University
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In a previous paper, we studied a general theory of integrally closed Henselian integrity domains and some properties of Henselian valuation rings. The present paper is its continuation. The main aim of the present paper is to study a general theory of Henselian local integrity domains in the present paper we call a ring o a local ring if o is a quasi-local ring and if the intersection of all powers of the maximal ideal of o is zero, and in this case we introduce a topology by taking the system of all powers of the maximal ideal as a system of neighbourhoods of zero.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 7 , June 1954 , pp. 1 - 19
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1954

References

1 On the theory of Henselian rings,- Nagoya Math. J. 5 (1953), pp. 45-57, which will be referred as [H.R.] in the present paper.

2 This lemma and the corollary were proved in Nagata, On the structure of complete local rings, Nagoya Math. J. 1 (1950).

3 This equality holds in general without assumption that o is a local ring.

4 In this case, the decomposition ring õ of over o is the decomposition ring of over o’ as will be easily seen from arguments in our proof in (2).

5 It will be not hard to see that

6 As was shown in [H.R.], there exists a valuation ring which is not algebraically closed in its completion. Some sufficient conditions for our problem will be shown in a latter paper.

7 Since that is an integrity domain is evident.

8 See Theorem 4 in Nagata, Some remarks on local rings, Nagoya Math. J. 6(1953).

9 Making no use of Theorem 6, we can prove this directly; see, G. Azumaya, On maximally central algebras, Nagoya Math. J. 2 (1950).

10 This means “modulo the ideal which is the intersection of q*with the ring of consideration.”

11 Some remarks on local rings II, Memo. Kyoto.

12It must be a generalized assertion also for the case of finite field. A proof of such generalization was given in a previous paper 1. c. note 8).

13 See, for inst., Krull, W., Dimensionstheorie in Stellenringen, J. Reine Angew. Math. 179 (1938)Google Scholar.

14 When an integrally closed quasi-local integrity domain o contains a field, then considering a finitely generated subring over a subfield of o, we can prove our results in the present paper, also without making no use of the lemma which we want to show now.