Free algebras and free groups in Ore extensions and free group algebras in division rings

JOURNAL OF ALGEBRA

卷:455

Article

Bell, Jason P.¹  Goncalves, Jairo Z.² 

[1] Univ Waterloo, Dept Pure Math, Waterloo, ON N2L 3G1, Canada

[2] Univ Sao Paulo, Dept Math, BR-05508090 Sao Paulo, Brazil

关键词: Division rings;    Free groups;    Free algebras;    Ore extensions;    Automorphisms;    Derivations;    Solvable groups;   

DOI  :  10.1016/j.jalgebra.2016.02.011

来源: Elsevier

【 摘 要 】

Let K be a field of characteristic zero, let sigma be an automorphism of K and let delta be a sigma-derivation of K. We show that the division ring D = K(x; sigma, delta) either has the property that every finitely generated subring satisfies a polynomial identity or D contains a free algebra on two generators over Its center. In the case when K is finitely generated over a subfield k we then see that for sigma a k-algebra automorphism of K and delta a k-linear derivation of K, K(x; sigma) having a free subalgebra on two generators is equivalent to sigma having infinite order, and K(x; delta) having a free subalgebra is equivalent to delta being nonzero. As an application, we show that if D is a division ring with center k of characteristic zero and D* contains a solvable subgroup that is not locally abelian-by-finite, then D contains a free k-algebra on two generators. Moreover, if we assume that k is uncountable, without any restrictions on the characteristic of k, then D contains the k-group algebra of the free group of rank two. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free