Limit T-subalgebras in free associative algebras
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- 作者：Dimas José ; Gonç ; alvesa ; 1 ; dimasjog@gmail.com" class="auth_mail ; Alexei Krasilnikovb ; 2 ; alexei@unb.br" class="auth_mail ; Irina Sviridovab ; 3 ; irina@mat.unb.br" class="auth_mail
- 关键词：16R10 ; 16R40 ; 16R99 ; 16S10 ; 8B20 ; 8A30
- 刊名：Journal of Algebra
- 出版年：15 August 2014
- 年：2014
- 卷：412
- 期：Complete
- 页码：264-280
- 全文大小：336 K
文摘
Let F銆圶銆?/span> be the free unitary associative algebra over a field F on a free generating set X . An unitary subalgebra R of F銆圶銆?/span> is called a T-subalgebra if R is closed under all endomorphisms of F銆圶銆?/span>. A T -subalgebra R鈦?/sup> in F銆圶銆?/span> is limit if every larger T -subalgebra W猥孯鈦?/sup> is finitely generated (as a T -subalgebra) but R鈦?/sup> itself is not. It follows easily from Zorn's lemma that if a T -subalgebra R is not finitely generated then it is contained in some limit T -subalgebra R鈦?/sup>. In this sense limit T -subalgebras form a “border” between those T -subalgebras which are finitely generated and those which are not. In the present article we give the first example of a limit T -subalgebra in F銆圶銆?/span>, where F is an infinite field of characteristic p>2 and |X|≥4. Note that, by Shchigolev's result, over a field F of characteristic 0 every T -subalgebra in F銆圶銆?/span> is finitely generated; hence, over such a field limit T -subalgebras in F銆圶銆?/span> do not exist.