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On free subsemigroups of associative algebras


Author: Edward S. Letzter
Journal: Proc. Amer. Math. Soc. 148 (2020), 457-460
MSC (2010): Primary 20M25, 16U99; Secondary 20M05
DOI: https://doi.org/10.1090/proc/14799
Published electronically: November 13, 2019
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Abstract: In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $ k$-algebras $ R$, where $ k$ is a field not algebraic over a finite subfield. We show that $ R$ contains a free noncyclic subsemigroup in the following cases: (1) $ R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $ R$ has at least one nonartinian primitive subquotient. (3) $ k$ is uncountable and $ R$ is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Klein's conjecture for numerous well-known classes of domains, over countable fields, not covered in the prior literature.


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Additional Information

Edward S. Letzter
Affiliation: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: letzter@temple.edu

DOI: https://doi.org/10.1090/proc/14799
Keywords: Subsemigroup, free semigroup, associative algebra
Received by editor(s): April 16, 2019
Published electronically: November 13, 2019
Communicated by: Sarah Witherspoon
Article copyright: © Copyright 2019 American Mathematical Society