Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T05:32:15.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher commutators, ideals and cardinality

Published online by Cambridge University Press:  17 April 2009

Charles Lanski
Charles Lanski
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, United States of America, e-mail: clanski@ mtha.usc.edu
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.

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 54 , Issue 1 , August 1996 , pp. 41 - 54
Copyright
Copyright © Australian Mathematical Society 1996

References

REFERENCES

[1]Herstein, I.N., Noncommutative rings, Carus Mathematical Monographs 15 (Mathematical Association of America, 1968).Google Scholar
[2]Herstein, I.N., Rings with involution (University of Chicago Press, Chicago, 1976).Google Scholar
[3]Hirano, Y., ‘Some finiteness conditions for rings’, Chinese J. Math. 16 (1988), 55–59.Google Scholar
[4]Hirano, Y., ‘On a problem of Szász’, Bull. Austral. Math. Soc. 40 (1989), 363–364.CrossRefGoogle Scholar
[5]Jacobson, N., PI-algebras, Lecture Notes in Mathematics 441 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).Google Scholar
[6]Kaplansky, I., Set theory and metric spaces (Chelsea Publishing Company, New York, 1977).Google Scholar
[7]Koh, K., ‘On properties of rings with a finite number of zero divisors’, Math. Ann. 171 (1976), 79–80.CrossRefGoogle Scholar
[8]Lanski, C., ‘Rings with few nilpotents’, Houston J. Math. 18 (1992), 577–590.Google Scholar
[9]Lanski, C., ‘On the cardinality of rings with special subsets which are finite’, Houston J. Math. 19 (1993), 357–373.Google Scholar
[10]Lanski, C. and Montgometry, S., ‘Lie structure of prime rings of characteristic 2’, Pacific J. Math. 42 (1972), 117–136.CrossRefGoogle Scholar
[11]Lewin, J., ‘Subrings of finite index in finitely generated rings’, J. Algebra 5 (1967), 84–88.CrossRefGoogle Scholar