Submission: 2000, May 19, revised: 2000, June 4
Let $D^*$ be the multiplicative group of a division ring $D$. We show that the multiplicative version of the Herstein's theorem  does not hold, namely $[D^*,[D^*,D^*]] \neq [D^*,D^*]$. This opens up an area to study a central series of $D^*$. We show that the structure of subgroups which appear in a central series of $D^*$ affect greatly the structure of $D^*$. For example it is shown that if every element of a subgroup in a central series of $D^*$ is algebraic over the center of $D$ then $D$ is algebraic division algebra. This generalize the recent results of algebracity of the commutator subgroup $[D^*,D^*]$ and its role on the structure of division ring. We then generalize the classical theorems of Kaplansky and Jacobson on commutativity of a division ring.
1991 Mathematics Subject Classification:
Keywords and Phrases: Division algebra. Descending central series, Valuation theory.
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