Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-01T00:43:28.414Z Has data issue: false hasContentIssue false
  • English
  • Français

On maximal nilpotent subrings of right Noetherian rings

Published online by Cambridge University Press:  18 May 2009

Gerhard Michler
Gerhard Michler
Affiliation:
Johann Wolfgang Goethe UniversitätFrankfurt
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.

Applying Hopkins's Theorem asserting that each unitary right Artinian ring is right Noetherian, G. Köthe and K. Shoda proved the following theorem (cf. Köthe [7], p. 360, Theorem 1 and p. 363, Theorem 5): If R is a unitary right Artinian ring, then the following statements hold:

(i) Each nilpotent subring of R is contained in a maximal nilpotent subring of R.

(ii) The intersection of all maximal nilpotent subrings of R is the maximal nilpotent twosided ideal of R.

(iii) All maximal nilpotent subrings of R are conjugate.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 8 , Issue 2 , July 1967 , pp. 89 - 101
Copyright
Copyright © Glasgow Mathematical Journal Trust 1967

References

1.Barnes, D. W., On the radical of a ring with minimum condition, J. Australian Math. Soc. 5 (1965), 234–236.CrossRefGoogle Scholar
2.Faith, C. and Utumi, Y., On Noetherian prime rings, Trans. Amer. Math. Soc. 114, (1965), 53–60.CrossRefGoogle Scholar
3.Fuchs, L., Abelian groups (Oxford, London, New York, Paris, 1960).Google Scholar
4.Goldie, A. W., Semi-prime rings with maximum condition, Proc. London Math. Soc. Ser. 3 10 (1960), 201–220.CrossRefGoogle Scholar
5.Herstein, I. N. and Small, L., Nil rings satisfying certain chain conditions, Canad. J. Math. 16 (1964), 771–776.CrossRefGoogle Scholar
6.Jacobson, N., Structure of rings, Amer Math. Soc. Colloquium Publications 37 (Providence, 1964).Google Scholar
7.Köthe, G., Über maximale nilpotente Unterringe und Nilringe, Math. Ann. 103 (1930), 359–363.CrossRefGoogle Scholar
8.Levitzki, L., Über nilpotente Unterringe, Math. Ann. 105 (1931), 620–627.CrossRefGoogle Scholar
9.Levy, L., Unique subdirect sums of prime rings, Trans. Amer. Math. Soc. 106 (1963), 64–76.CrossRefGoogle Scholar
10.Michler, G., Klassische Quotientenringe von nicht notwendig endlich dimensionalen, halbprimen Ringen, Math. Z. (to appear).Google Scholar
11.Michler, G., Radikale und Sockel, accepted for publication by Math. Ann.Google Scholar
12.Procesi, C. and Small, L., On a theorem of Goldie, J. Algebra 2 (1965), 80–84.CrossRefGoogle Scholar