Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T06:59:28.105Z Has data issue: false hasContentIssue false
  • English
  • Français

Regular bisimple rings

Published online by Cambridge University Press:  20 January 2009

John Hannah
John Hannah
Affiliation:
Mathematics DepartmentUniversity Of CanterburyChristchurch, New Zealand
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.

We characterize regular bisimple rings in terms of some perspectivity conditions on their lattices of principal right ideals. We also show that, if S is the multiplicative subsemigroup generated by all the idempotents of a regular bisimple ring R, then

(i) if R does not have an identity, then S = R and has depth 2;

(ii) if R does have an identity but is not a division ring, then S = {a∈R:a is neither left nor right invertible} ∪ {1} and has depth 3.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 34 , Issue 1 , February 1991 , pp. 89 - 97
Copyright
Copyright © Edinburgh Mathematical Society 1991

References

REFERENCES

1.Goodearl, K. R., Von Neumann Regular Rings (Pitman, 1979).Google Scholar
2.Goodearl, K. R. and Handelman, D., Simple self-injective rings, Comm. Algebra 3 (1975), 797–834.Google Scholar
3.Hannah, J. and O'meara, K. C., Products of idempotents in regular rings II, J. Algebra 123 (1989), 223–239.CrossRefGoogle Scholar
4.Hannah, J. and O'meara, K. C., Depth of idempotent-generated subsemigroups of a regular ring, Proc. London Math. Soc. 59 (1989), 464–482.CrossRefGoogle Scholar
5.Howie, J. M., An Introduction to Semigroup Theory (Academic Press, 1976).Google Scholar
6.Howie, J. M., Products of idempotents in finite full transformation semigroups, Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 243–254.CrossRefGoogle Scholar
7.Jacobson, N., Structure of Rings (American Mathematical Society, 1964).Google Scholar
8.Lawrence, J., A countable self-injective ring is quasi-Frobenius, Proc. Amer. Math. Soc. 65 (1977), 217–220.Google Scholar
9.Munn, W. D., Bisimple rings, Quart. J. Math. Oxford 32 (1981), 181–191.CrossRefGoogle Scholar
10.O'meara, K. C., Products of idempotents in regular rings, Glasgow Math. J. 28 (1986), 143–152.CrossRefGoogle Scholar
11.Von Neumann, J., Continuous Geometry (Princeton University Press, 1960).Google Scholar