Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T00:52:39.641Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Theorem of Burgess and Stephenson

Part of: Local rings and generalizations Modules, bimodules and ideals

Published online by Cambridge University Press:  06 December 2018

W. K. Nicholson
Rights & Permissions [Opens in a new window]

Abstract

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.

A theorem of Burgess and Stephenson asserts that in an exchange ring with central idempotents, every maximal left ideal is also a right ideal. The proof uses sheaf-theoretic techniques. In this paper, we give a short elementary proof of this important theorem.

Keywords

exchange ringabelian ringleft quasi-duo ring

MSC classification

Primary: 16L60: Quasi-Frobenius rings
Secondary: 16D25: Ideals 16D60: Simple and semisimple modules, primitive rings and ideals
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 603 - 605
Copyright
© Canadian Mathematical Society 2018 

References

Burgess, W. and Stephenson, W., Rings all of whose Pierce stalks are local . Canad. Math. Bull. 22(1979), 159–164. https://doi.org/10.4153/CMB-1979-022-8.Google Scholar
Crawley, P. and Jónsson, B., Refinements for infinite direct decompositions of algebraic systems . Pacific J. Math. 14(1964), 797–855.Google Scholar
Feller, E. H., Properties of primary noncommutative rings . Trans. Amer. Math. Soc. 89(1958), 79–91. https://doi.org/10.2307/1993133.Google Scholar
Huh, C., Jang, S. H., Kim, C. O., and Lee, Y., Rings whose maximal one-sided ideals are two-sided . Bull. Korean Math. Soc. 39(2002), 411–422. https://doi.org/10.4134/BKMS.2002.39.3.411.Google Scholar
Lam, T. Y. and Dugas, A. S., Quasi-duo rings and stable range descent . J. Pure and Appl. Algebra 195(2005), 243–259. https://doi.org/10.1016/j.jpaa.2004.08.011.Google Scholar
Nicholson, W. K., Lifting idempotents and exchange rings . Trans. Amer. Math. Soc. 229(1977), 269–278. https://doi.org/10.2307/1998510.Google Scholar
Warfield, R., Exchange rings and decompositions of modules . Math. Ann. 199(1972), 31–36. https://doi.org/10.1007/BF01419573.Google Scholar
Yu, H.-P., On quasi-duo rings . Glasgow Math. J. 37(1995), 21–31. https://doi.org/10.1017/S0017089500030342.Google Scholar