Hostname: page-component-76dd75c94c-ccc76 Total loading time: 0 Render date: 2024-04-30T08:34:03.958Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Studies on Semi-local Rings

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Mathematical Institute, Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The concept of semi-local rings was introduced by C. Chevalley [1], which the writer has generalized in a recent paper [7] by removing the chain condition. The present paper aims mainly at the study of completions of semi-local rings. First in § 1 we investigate semi-local rings which are subdirect sums of semi-local rings, and we see in § 2 that a Noetherian semi-local ring R is complete if (and only if) R/p is complete for every minimal prime divisor p of zero ideal, together with some other properties. Further we consider in § 3 subrings of the completion of a semi-local ring. § 4 gives some supplementary remarks to [7], Chapter II, Proposition 8.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 3 , October 1951 , pp. 23 - 30
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

[1] Chevalley, C., On the theory of local rings, Ann. Math. 44, pp. 690–708 (1943).Google Scholar
[2] Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc. 59, pp. 54–106 (1946).CrossRefGoogle Scholar
[3] Cohen, I. S., Commutative rings with restricted minimum condition, Duke Math, J. 17, pp. 27–42 (1950).CrossRefGoogle Scholar
[4] Grell, H., Beziehungen zwissen den Idealen verschiedener Ringe, Math. Ann. 97, pp. 409–523 (1927).CrossRefGoogle Scholar
[5] Mori, S., Allgemeine Z.P.I-Ringe, J. Sci. Hirosima Univ. A, 10, pp. 117–136 (1940).Google Scholar
[6] Nagata, M., On the structure of complete local rings, Nagoya Math. J. 1, pp. 63–70 (1950).CrossRefGoogle Scholar
[7] Nagata, M., On the theory of semi-local ring, forthcoming.Google Scholar
[8] Nagata, M., On the theory of radicals in a ring, forthcoming.Google Scholar
[9] Nagata, M., Note on subdirect sums of rings, Nagoya Math. J. 2, pp. 49–53 (1951).Google Scholar
[10] Uzkov, A. I., On the rings of quotients of commutative rings, Mat. Sbornik N. S. 22 (64) (1948) [in Russian].Google Scholar
[11] Zariski, O., Generalized semi-local rings, Sum. Bras. Math. 1, pp. 169–195 (1946).Google Scholar