Limit T-subalgebras in free associative algebras

JOURNAL OF ALGEBRA

卷:412

Article

Goncalves, Dimas Jose¹  Krasilnikov, Alexei²  Sviridova, Irina² 

[1] Univ Fed Sao Carlos, Dept Matemat, BR-13565905 Sao Carlos, SP, Brazil

[2] Univ Brasilia, Dept Matemat, BR-70910900 Brasilia, DF, Brazil

关键词: Free associative algebra;    Polynomial identity;    Central polynomial;    T-ideal;    T-subspace;   

DOI  :  10.1016/j.jalgebra.2014.03.032

来源: Elsevier

【 摘 要 】

Let F < X > be the free unitary associative algebra over a field F on a free generating set X. An unitary subalgebra R of F < X > is called a T-subalgebra if R is closed under all endomorphisms of F < X >. A T-subalgebra R* in F < X > 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'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 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 < X >, where F is an infinite field of characteristic p > 2 and vertical bar X vertical bar >= 4. Note that, by Shchigolev's result, over a field F of characteristic 0 every T-subalgebra in F < X > is finitely generated; hence, over such a field limit T-subalgebras in F < X > do not exist. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free