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Paper IPM / M / 41  


Abstract:  
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
Fcentral division ring if and only if all its additive
commutators are of bounded degree over F. Furthermore, we study
the Fvector space D/[D,D] and show that dim_{F} 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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