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

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• 作者：Jason P. Bell^a ; ^{jpbell@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author} ; Jairo Z. Gonç ; alves^b ; ^{jz.goncalves@usp.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16K40 ; 16A06 ; 14J50
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：455
• 期：Complete
• 页码：235-250
• 全文大小：403 K

文摘

Let K be a field of characteristic zero, let σ be an automorphism of K and let δ be a σ-derivation of K  . We show that the division ring class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316001071&_mathId=si1.gif&_user=111111111&_pii=S0021869316001071&_rdoc=1&_issn=00218693&md5=c4dacbbbb78c2bcdbb71a1638557571d" title="Click to view the MathML source">D=K(x;σ,δ)class="mathContainer hidden">class="mathCode">$D=K(x;\sigma ,\delta )$ 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 σ a k-algebra automorphism of K and δ a k-linear derivation of K  , class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316001071&_mathId=si2.gif&_user=111111111&_pii=S0021869316001071&_rdoc=1&_issn=00218693&md5=7d2a1cb19639c78e8032212ee9a31875" title="Click to view the MathML source">K(x;σ)class="mathContainer hidden">class="mathCode">$K(x;\sigma )$ having a free subalgebra on two generators is equivalent to σ   having infinite order, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316001071&_mathId=si24.gif&_user=111111111&_pii=S0021869316001071&_rdoc=1&_issn=00218693&md5=4066d81f6e7e24aa70d8dbd7c9a4d324" title="Click to view the MathML source">K(x;δ)class="mathContainer hidden">class="mathCode">$K(x;\delta )$ having a free subalgebra is equivalent to δ being nonzero. As an application, we show that if D is a division ring with center k   of characteristic zero and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316001071&_mathId=si4.gif&_user=111111111&_pii=S0021869316001071&_rdoc=1&_issn=00218693&md5=ef9351e84c96a2f5ac2dbc8c8eaed86d" title="Click to view the MathML source">D^⁎class="mathContainer hidden">class="mathCode">$D^{⁎}$ 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.