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Commutator rings

Published online by Cambridge University Press:  17 April 2009

Zachary Mesyan
Zachary Mesyan
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, United States of America e-mail: [email protected]
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A ring is called a commutator ring if every element is a sum of additive commutators. In this note we give examples of such rings. In particular, we show that given any ring R, a right R-module N, and a nonempty set Ω, EndR(⌖ΩN) and EndR(ΠΩN) are commutator rings if and only if either Ω is infinite or EndR(N) is itself a commutator ring. We also prove that over any ring, a matrix having trace zero can be expressed as a sum of two commutators.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 2 , October 2006 , pp. 279 - 288
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Albert, A.A. and Muckenhoupt, B., ‘On matrices of trace zero’, Michigan Math. J. 4 (1957), 1–3.CrossRefGoogle Scholar
[2]Amitsur, S.A. and Rowen, L.H., ‘Elements of reduced trace 0’, Israel J. Math. 87 (1994), 161–179.CrossRefGoogle Scholar
[3]Dixmier, J., ‘Sur les algèbres de Weyl’, Bull. Soc. Math. France 96 (1968), 209–242.CrossRefGoogle Scholar
[4]Harris, B., ‘Commutators in division rings’, Proc. Amer. Math. Soc. 9 (1958), 628–630.CrossRefGoogle Scholar
[5]Kaplansky, I., ‘“Problems in the theory of rings” revisited’, Amer. Math. Monthly 77 (1970), 445–454.Google Scholar
[6]Lazerson, E., ‘Onto inner derivations in division rings’, Bull. Amer. Math. Soc. 67 (1961), 356–358.CrossRefGoogle Scholar
[7]Revoy, P., ‘Algèbres de Weyl en caractéistique p’, C. R. Acad. Set. Paris Sr. A-B 276 (1973), A225–A228.Google Scholar
[8]Rosset, M., Elements of trace zero and commutators, (Ph.D. Thesis) (Bar-Ilan University, 1997).Google Scholar
[9]Rosset, M. and Rosset, S., ‘Elements of trace zero in central simple algebras’, in Rings, Extensions and Cohomology, (Magid, A., Editor) (Marcel Dekker, 1994), pp. 205–215.Google Scholar
[10]Rosset, M. and Rosset, S., ‘Elements of trace zero that are not commutators’, Comm. Algebra 28 (2000), 3059–3072.CrossRefGoogle Scholar
[11]Shoda, K., ‘Einige Sätze über Matrizen’, Japan J. Math. 13 (1936), 361–365.CrossRefGoogle Scholar