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 a0 + a1x + ⋯ + anxn (resp. right polynomial a0 + xa1 + ⋯ + xnan) over K such that a0 + a1a + ⋯ + anan = 0 (resp. a0 + aa1 + ⋯ + anan). 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.
Highlights
A theorem due to Jacobson [7] says that every algebraic division algebra of bounded degree over its center, is centrally finite
Bell et al in [4] using a technique based on combinatorics of words have provided a generalization of Jacobson Theorem from other aspects of view. They proved that every division algebra whose elements are algebraic of bounded degree over a subfield, must be centrally finite. They proved that if D is a division algebra with center F such that its elements are left algebraic of bounded degree n over a subfield K, dimF D ≤ n2
We prove that a division algebra whose all multiplicative commutators are left algebraic of bounded degree over a subfield is centrally finite, provided that has infinite center
Summary
A theorem due to Jacobson [7] says that every algebraic division algebra of bounded degree over its center, is centrally finite. Bell et al in [4] using a technique based on combinatorics of words have provided a generalization of Jacobson Theorem from other aspects of view They proved that every division algebra whose elements are algebraic of bounded degree over a (not necessarily central) subfield, must be centrally finite. We prove that a division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield is centrally finite, provided that has infinite center. As another result we show that if D is a division algebra with center F whose multiplicative group of commutators is left (right) algebraic of bounded degree n over a (not necessarily central) subfield, dimF D ≤ n2. Among other results we prove that if D is a division algebra with infinite center F and non-central subfield K with a ∈ K \ F algebraic over F such that for every x ∈ D, ax − xa is left (right) algebraic over subfield K of bounded degree n, D is centrally finite
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