Limit T-subalgebras in free associative algebras

作者：Goncalves Dimas Jose^*; Krasilnikov Alexei; Sviridova Irina

来源：Journal of Algebra, 2014, 412: 264-280.

DOI：10.1016/j.jalgebra.2014.03.032

• Free associative algebra
• Polynomial identity
• Central polynomial
• T-ideal
• T-subspace

Let F %26lt; X %26gt; be the free unitary associative algebra over a field F on a free generating set X. An unitary subalgebra R of F %26lt; X %26gt; is called a T-subalgebra if R is closed under all endomorphisms of F %26lt; X %26gt;. A T-subalgebra R* in F %26lt; X %26gt; is limit if every larger T-subalgebra W not superset of R* is finitely generated (as a T-subalgebra) but R* itself is not. It follows easily from Zorn%26apos;s lemma that if a T-subalgebra R is not finitely generated then it is contained in some limit T-subalgebra R*. In this sense limit T-subalgebras form a %26quot;border%26quot; 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 %26lt; X %26gt;, where F is an infinite field of characteristic p %26gt; 2 and vertical bar X vertical bar %26gt;= 4. Note that, by Shchigolev%26apos;s result, over a field F of characteristic 0 every T-subalgebra in F %26lt; X %26gt; is finitely generated; hence, over such a field limit T-subalgebras in F %26lt; X %26gt; do not exist.

• 出版日期2014-8-15