arXiv subjects
Publications and source records attributed to Mehdi Aaghabali.
Let D be a division ring finite dimensional over its center F. The goal of this paper is to prove that for any positive integer n there exists a in D^(n); the n-th multiplicative derived subgroup, such that F(a) is a maximal subfield of D. We also show that a single depth-n iterated additive commutator would generate a maximal subfield of D.
Let D be a division algebra with center F. A maximal subfield of D is defined to be a field K such that CD(K) = K; that is, K is its own centralizer in D. A maximal subfield K is said to be self-invariant if it normalises by itself, i.e. ND*(K)= K: This kind of subfields is important because they have strong connexion with the most famous Albert's Conjecture (every division ring of prime index is cyclic). In fact, we pose a question that asserts whether every division ring whose all maximal subfields are self-invariant has to be commutative. The positive answer to this question, in finite dimensional case, implies the Albert's Conjecture (see x2). Although we show the Mal'cev-Neumann division ring demonstrates negative answer in the case of infinite dimensional division rings, but it is still most likely the question receives positive answer if we restrict ourselves to the finite dimensional division rings. We also have had the opportunity to use the Mal'cev-Neumann structure to answer Conjecture 1 below in negative (see x3). Finally, among other things, we rely on this kind of subfields to present a criteria for a division ring to have finite dimensional subdivision ring (see x4).
Let D be a division ring with centre F. 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.