Remarks on vector space generated by the multiplicative commutators of a division ring
DOI:
https://doi.org/10.7146/math.scand.a-116324Abstract
Let D be a division ring with centre F. An element of the form xyx−1y−1∈D is called a multiplicative commutator. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In M. Aghabali et al., J. Algebra Appl. 12 (2013), no. 8, art. 1350043, the authors have conjectured that every division ring is generated as a vector space over its centre by all of its multiplicative commutators. In this note it is shown that if D is centrally finite, then the conjecture holds.
References
Aghabali, M., Akbari, S., Ariannejad, M., and Madadi, A., Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12 (2013), no. 8, art. 1350043, 7 pp. https://doi.org/10.1142/S0219498813500436
Akbari, S., Arian-Nejad, M., and Mehrabadi, M. L., On additive commutator groups in division rings, Results Math. 33 (1998), no. 1-2, 9–21. https://doi.org/10.1007/BF03322065
Hazrat, R., A note on multiplicative commutators of division rings, J. Algebra Appl. 18 (2019), no. 2, art. 1950031, 2 pp. https://doi.org/10.1142/S0219498819500312
Herstein, I. N., Topics in ring theory, The University of Chicago Press, Chicago, Ill.-London, 1969.
Lam, T. Y., A first course in noncommutative rings, second ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4419-8616-0
Morandi, P., Field and Galois theory, Graduate Texts in Mathematics, vol. 167, Springer-Verlag, New York, 1996. https://doi.org/10.1007/978-1-4612-4040-2
Rowen, L. H., Ring theory, student ed., Academic Press, Inc., Boston, MA, 1991.
Akbari, S., Arian-Nejad, M., and Mehrabadi, M. L., On additive commutator groups in division rings, Results Math. 33 (1998), no. 1-2, 9–21. https://doi.org/10.1007/BF03322065
Hazrat, R., A note on multiplicative commutators of division rings, J. Algebra Appl. 18 (2019), no. 2, art. 1950031, 2 pp. https://doi.org/10.1142/S0219498819500312
Herstein, I. N., Topics in ring theory, The University of Chicago Press, Chicago, Ill.-London, 1969.
Lam, T. Y., A first course in noncommutative rings, second ed., Graduate Texts in Mathematics, vol. 131, Springer-Verlag, New York, 2001. https://doi.org/10.1007/978-1-4419-8616-0
Morandi, P., Field and Galois theory, Graduate Texts in Mathematics, vol. 167, Springer-Verlag, New York, 1996. https://doi.org/10.1007/978-1-4612-4040-2
Rowen, L. H., Ring theory, student ed., Academic Press, Inc., Boston, MA, 1991.
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Published
2020-05-06
How to Cite
Aaghabali, M., & Tajfirouz, Z. (2020). Remarks on vector space generated by the multiplicative commutators of a division ring. MATHEMATICA SCANDINAVICA, 126(2), 161–164. https://doi.org/10.7146/math.scand.a-116324
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