Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-13T08:12:23.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Rings with Enough Invertible Ideals

Published online by Cambridge University Press:  20 November 2018

P. F. Smith
P. F. Smith*
Affiliation:
University of Glasgow, Glasgow, Scotland
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.

All rings are associative with identity element 1 and all modules are unital. A ring has enough invertible ideals if every ideal containing a regular element contains an invertible ideal. Lenagan [8, Theorem 3.3] has shown that right bounded hereditary Noetherian prime rings have enough invertible ideals. The proof is quite ingenious and involves the theory of cycles developed by Eisenbud and Robson in [5] and a theorem which shows that any ring S such that R ⊆ S ⊆ Q satisfies the right restricted minimum condition, where Q is the classical quotient ring of R. In Section 1 we give an elementary proof of Lenagan's theorem based on another result of Eisenbud and Robson, namely every ideal of a hereditary Noetherian prime ring can be expressed as the product of an invertible ideal and an eventually idempotent ideal (see [5, Theorem 4.2]). We also take the opportunity to weaken the conditions on the ring R.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 1 , 01 February 1983 , pp. 131 - 144
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Amitsur, S. A. and Small, L. W., Polynomials over division rings, Israel J. Math. 81 (1978), 353–358.Google Scholar
2. Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer-Verlag, 1974).Google Scholar
3. Brown, K. A., Primitive group rings of soluble groups, preprint.Google Scholar
4. Eisenbud, D. and Robson, J. C., Modules over Dedekind prime rings, J. Algebra 16 (1970), 67–85.Google Scholar
5. Eisenbud, D. and Robson, J. C., Hereditary Noetherian prime rings, J. Algebra 16 (1970), 86–104.Google Scholar
6. Goldie, A. W., Semiprime rings with maximum condition, Proc. London Math. Soc. (3) 10 (1960), 201–220.Google Scholar
7. Lenagan, T. H., Bounded Asano orders are hereditary, Bull. London Math. Soc. 8 (1971), 67–69.Google Scholar
8. Lenagan, T. H., Bounded hereditary Noetherian prime rings, J. London Math. Soc. (2) 6 (1973), 241–246.Google Scholar
9. Passman, D. S., The algebraic structure of group rings (Wiley, 1977).Google Scholar
10. Roseblade, J. E. and Smith, P. F., A note on hypercentral group rings, J. London Math. Soc. (2) 13 (1976), 183–190.Google Scholar
11. Small, L. W., Semi-hereditary rings, Bull. Amer. Math. Soc. 78 (1967), 656–658.Google Scholar
12. Webber, D. B., Ideals and modules of simple Noetherian hereditary rings, J. Algebra 16 (1970), 239–242.Google Scholar