Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T08:43:42.991Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the invertible ideal theorem

Published online by Cambridge University Press:  18 May 2009

Andy J. Gray
Andy J. Gray
Affiliation:
Mathematics Institute, University of Warwick, Coventry Cv4 7Al
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.

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 1 , January 1984, pp. 27 - 30
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Chatters, A. W., Goldie, A. W., Hajarnavis, C. R. and Lenagan, T. H., Reduced rank in Noetherian rings, J. Algebra 61 (1979), 582–589.CrossRefGoogle Scholar
2.Deshpande, V. K., Completions of Noetherian hereditary prime rings, Pacific J. Math. 90 (1980), 285–297.CrossRefGoogle Scholar
3.Eisenbud, D. and Robson, J. C., Hereditary Noetherian prime rings, J. Algebra 16 (1970), 86–104.CrossRefGoogle Scholar
4.Gwynne, W. D. and Robson, J. C., Completions of non-commutative Dedekind prime rings, J. London Math. Soc (2) 4 (1971), 346–352.CrossRefGoogle Scholar
5.Jategaonkar, A. V., Relative Krull dimension and prime ideals in right Noetherian rings, Comm. Algebra 4 (1974), 429–468.Google Scholar
6.Jategaonkar, A. V., Principal Ideal Theorem for Noetherian P. I. rings, J. Algebra 35 (1975), 17–22.CrossRefGoogle Scholar
7.Jategaonkar, A. V., Relative Krull dimension and prime ideals in right Noetherian rings: An addendum, Comm. Algebra 10 (1982), 361–366.CrossRefGoogle Scholar
8.Kaplansky, I., Commutative rings (Allyn and Bacon, 1970).Google Scholar