Groups and noncommutative algebra: interactions and applications
On the unit group of Z-orders in finite dimensional algebras
| Full text | |
| Author(s): |
Total Authors: 2
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| Affiliation: | [1] Univ Waterloo, Dept Pure Math, Waterloo, ON N2L 3G1 - Canada
[2] Univ Sao Paulo, Dept Math, BR-05508090 Sao Paulo - Brazil
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | Journal of Algebra; v. 455, p. 235-250, JUN 1 2016. |
| Web of Science Citations: | 2 |
| Abstract | |
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) | |
| FAPESP's process: | 09/52665-0 - Groups, rings and algebras: interactions and applications |
| Grantee: | Francisco Cesar Polcino Milies |
| Support Opportunities: | Research Projects - Thematic Grants |