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

Autor(es)

Bell, Jason P. ^[1] ; Goncalves, Jairo Z. ^[2]

Afiliação do(s) autor(es)

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

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

• Tipo de documento: Artigo Científico
• Fonte: Journal of Algebra; v. 455, p. 235-250, JUN 1 2016.
• Citações Web of Science: 2

Resumo

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. (AU)

• Processo FAPESP: 09/52665-0 - Grupos, anéis e álgebras: interações e aplicações
• Beneficiário: Francisco Cesar Polcino Milies
• Modalidade de apoio: Auxílio à Pesquisa - Temático