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Quasi-coefficient rings of a local ring

Published online by Cambridge University Press:  22 January 2016


Hideyuki Matsumura
Affiliation:
Department of Mathematics Nagoya University

Extract

In this note we will make a few observations on the structure of fields and local rings. The main point is to show that a weaker version of Cohen structure theorem for complete local rings holds for any (not necessarily complete) local ring. The consideration of non-complete case makes the meaning of Cohen’s theorem itself clearer. Moreover, quasi-coefficient fields (or rings) are handy when we consider derivations of a local ring.


Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1977

References

[1] Bourbaki, N., Algebre Commutative, Ch. 3,4. Hermann, Paris, 1961.Google Scholar
[2] Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59 (1946), 54106.Google Scholar
[3] Grothendieck, A. and Dieudonne, J., Elements de Geometrie Algebrique, Ch. IV, Premiere Partie, Publ. IHES, No. 20, 1964.Google Scholar
[4] Matsumura, H., Commutative Algebra, Benjamin, New York 1970.Google Scholar
[5] Nagata, M., Local Rings, Interscience, New York 1962.Google Scholar
[6] Nagata, M., Note on coefficient fields of complete local rings. Mem. Coll. Sci., Univ. Kyoto 32 (1959), 9192.Google Scholar
[7] Nagata, M., Note on complete local integrity domains. Mem. Coll. Sci., Univ. Kyoto 28 (1954), 271278.Google Scholar
[8] Chevalley, C., Some properties of ideals in rings of power series. Trans. Amer. Math. Soc. 55 (1944), 6884.Google Scholar

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