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Some Remarks on Local Rings

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Mathematical Institute, Nagoya University
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Previously C. Chevalley [1] proved the followings:

1. Let x1,…, Xn be algebraically independent elements over a field t which has infinitely many elements. Then :

a) If y is an element of [x1,…, xn] and if y is not in , then there exist elements y2…, yn of [x1,…, xn] such that [x1,…, xn] is integral over [y1, y2,…, yn]

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 6 , October 1953 , pp. 53 - 58
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1953

References

[1] Chevalley, C., Intersections of algebraic and algebroid varieties, Trans. Amer. Math. Soc, 57 (1945), pp. 1–85.CrossRefGoogle Scholar
[2] Nagata, M., Some studies on semi-local rings, Nagoya Math. J., 3 (1951), pp. 23–30.CrossRefGoogle Scholar
[3] Samuel, P., Sur les variétés algebroïdes, Ann. Inst. Fourier, 2 (1950), pp. 147–160.CrossRefGoogle Scholar
[4] Zariski, O., Analytical irreducibility of normal varieties, Ann. of Math., 49 (1948), pp. 352–361.CrossRefGoogle Scholar
[5] Zariski, O., Sur la normalité analytique des variété normales, Ann. Inst. Fourier, 2 (1950), pp. 161–164.CrossRefGoogle Scholar
[6] Zariski, O., Foundations of a general theory of birational correspondences, Trans. Amer. Math. Soc, 53 (1943), pp. 490–542.CrossRefGoogle Scholar