Roozbeh Hazrat: On the central series of the multiplicative group of division rings

rhazrat@Mathematik.Uni-Bielefeld.DE

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 [3] 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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