Vector Space Generated by the Multiplicative Commutators of a Division Ring

Abstract

Let D be a division ring with centre F. An element of the form xyx −1 y −1 ∈ D is called a multiplicative commutator. Let T (D) be the vector space over F generated by all multiplicative commutators in D. In [1], 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.

A note on multiplicative commutators of division rings

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We give an example of a division ring D whose multiplicative commutator subgroup does not generate D as a vector space over its centre, thus disproving the conjecture posed in [1].

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“…There are classical results due to Herstein, Kaplansky and Scott, among others, showing that the group D ′ is "dense" in D (see for example [3, §13]). In [1] the authors study the F -vector space T (D) generated by the set of multiplicative commutators. They prove that if T (D) is radical over F , then D = F , and if dim F T (D) < ∞, then dim F D < ∞.…”

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

“…There are wild studies on certain substructures of a division ring including a lot of conjectures on the structural properties arising from these substructures (see [2,3] and references therein). For a given division ring considering an special property P, say commutativity, algebraicity or other finiteness conditions, E-mail Addresses: a mehdi.aaghabali@ut.ac.ir, maghabali@gmail.com, b mhbien@hcmus.edu.vn The authors would like to thank Mehran Motiee for his fruitful discussions and helpful comments.…”