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Commutator rings

  • Zachary Mesyan (a1)

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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 EndRΩN) 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.

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References

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[1]Albert, A.A. and Muckenhoupt, B., ‘On matrices of trace zero’, Michigan Math. J. 4 (1957), 13.
[2]Amitsur, S.A. and Rowen, L.H., ‘Elements of reduced trace 0’, Israel J. Math. 87 (1994), 161179.
[3]Dixmier, J., ‘Sur les algèbres de Weyl’, Bull. Soc. Math. France 96 (1968), 209242.
[4]Harris, B., ‘Commutators in division rings’, Proc. Amer. Math. Soc. 9 (1958), 628630.
[5]Kaplansky, I., ‘“Problems in the theory of rings” revisited’, Amer. Math. Monthly 77 (1970), 445454.
[6]Lazerson, E., ‘Onto inner derivations in division rings’, Bull. Amer. Math. Soc. 67 (1961), 356358.
[7]Revoy, P., ‘Algèbres de Weyl en caractéistique p’, C. R. Acad. Set. Paris Sr. A-B 276 (1973), A225A228.
[8]Rosset, M., Elements of trace zero and commutators, (Ph.D. Thesis) (Bar-Ilan University, 1997).
[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. 205215.
[10]Rosset, M. and Rosset, S., ‘Elements of trace zero that are not commutators’, Comm. Algebra 28 (2000), 30593072.
[11]Shoda, K., ‘Einige Sätze über Matrizen’, Japan J. Math. 13 (1936), 361365.
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Commutator rings

  • Zachary Mesyan (a1)

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