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Paper   IPM / M / 41
School of Mathematics
  Title:   On additive commutator groups in division rings
1.  M. Arian-Nejad
2.  S. Akbari
3.  M. L. Mehrabadi
  Status:   Published
  Journal: Results Math.
  Vol.:  33
  Year:  1998
  Pages:   9-21
  Supported by:  IPM
Let D be a division ring with center F and denote by [D,D] the group generated additively by additive commutators. First, it is shown that in zero characteristic, D is algebraic over F if and only if each element of [D,D] is algebraic over F. We conjecture this assertion is true for any characteristic. Also, as a generalization of Jacobson's Theorem it is proved that D is an F-central division ring if and only if all its additive commutators are of bounded degree over F. Furthermore, we study the F-vector space D/[D,D] and show that dimF D/ [D,D] ≤ 1 if D is algebraic over F and char F=0. We then prove that any algebraic division ring contains a separable additive commutator over F except in one special case. Finally, the existence of primitive elements in [D,D] is studied for finite separable extensions of F in D.

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