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On the Structure of Complete Local Rings

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Nagoya University
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The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ring R (we say that a commutative ring R is Noetherian if every ideal in R has a finite basis and if R contains the identity) which has only one maximal ideal m. If the powers of m are defined as a system of neighbourhoods of zero, then R becomes a topological ring satisfying the first axiom of countability, And the notion was studied recently by C. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 1 , June 1950 , pp. 63 - 70
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1950

References

[1] Cohen, I. S.; On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. Vol. 59, No. 1, pp. 54–106 (1946).CrossRefGoogle Scholar
[2] Krull, W.; Dimensionstheorie in Stellenringen, J. Reine Angew. Math. Vol. 179, pp. 204–226 (1938).CrossRefGoogle Scholar
[3] Chevalley, C.; On the theory of local rings, Ann. of Math. Vol. 44, pp. 690–708 (1943).CrossRefGoogle Scholar
[4] Teichmüller, O.; Über die Struktur diskrete bewerteter perfect Körper, Ges. d. Wiss. Nachrichten Math. -Phys. Kl. Fachgr. I.N.F. Vol. 1, No. 10, pp. 151–161 (1936).Google Scholar
[5] Nagata, M.; On the theory of semi-local rings, forthcoming.Google Scholar