Division Algebras with Left Algebraic Commutators

Abstract

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a 0 + a 1 x + · · · + a n x n (resp. right polynomial a 0 +xa 1 +· · ·+x n a n ) over K such that a 0 +a 1 a+· · ·+a n a n = 0 (resp. a 0 +aa 1 +· · ·+a n a n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.

Certain simple maximal subfields in division rings

“…In the case n = 1, both results have been proved Chebotar et al in [6,Theorem 3,theorem 6], and recently again by the authors and S. Akbari in [1, Theorem 6, Theorem 7]. Both [1,6] and the current paper use rational polynomial identities in proving the aforementioned results. The idea is simple and clever: The key is a certain (non-commutative) polynomial g n (x, y 1 , .…”

“…The following result is the goal of this section. Note that the same result as Theorem 7, however, follows from the case n = 1, proved in [1] and [6], since by a theorem of Amitsur and Rowen D 1 = D 2 = D 3 = . .…”

“…First note that if charF = 0, then by a result due to Amitsur and Rowen [4] we have D 1 = D 2 = · · · . Hence, in this case the result follows from [1,Theorem 7]. In the case of charF = p > 0, the proof is similar to the one of Theorem 7.…”

“…, y n ) that vanishes E-mail Addresses: maghabali@gmail.com, mhbien@hcmus.edu.vn The first author's current institute: School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran whenever an algebraic element of degree ≤ n is substituted into x. One takes n < degD, substitutes a relevant rational expression into x and proves that the resulting expression cannot vanish on D ⊗ F L, where L is some splitting field of D. In [1] and [6], the expressions substituted into x are single additive, resp. multiplicative, commutators on two variables, whereas here, iterated commutators are considered.…”

Vành chia đại số trên vành chia con

“…Khái niệm đại số trên vành chia con đã từng được đề cập trong ( [1], Chương 7) nhằm nghiên cứu nghiệm của đa thức trên vành chia. Riêng lớp vành chia đại số trên vành chia con của nó cũng được nghiên cứu bởi nhiều nhà toán học lớn như I. N. Herstein và C. Faith (xem, chẳng hạn trong [2], [3]) và gần đây nó nhận được sự quan tâm nhiều hơn (xem [4]- [7]). Một trong những vấn đề mà ta quan tâm là khái niệm đại số phải và đại số trái có trùng nhau không?…”