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 $xyx^{-1}y^{-1}\in D$ 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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