Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-22T02:31:10.976Z Has data issue: false hasContentIssue false
  • English
  • Français

On free subalgebras of varieties

Part of: Varieties General nonassociative rings

Published online by Cambridge University Press:  22 November 2021

Renato Fehlberg Júnior Open the ORCID record for Renato Fehlberg Júnior [Opens in a new window]  and
Javier Sánchez Open the ORCID record for Javier Sánchez [Opens in a new window]
Renato Fehlberg Júnior
Affiliation:
Department of Mathematics, Universidade Federal do Espírito Santo, Vitoria, Espírito Santo, Brazil ([email protected])
Javier Sánchez
Affiliation:
Department of Mathematics, IME, University of São Paulo, São Paulo, SP, 05508-090, Brazil ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that some results of L. Makar-Limanov, P. Malcolmson and Z. Reichstein on the existence of free-associative algebras are valid in the more general context of varieties of algebras.

Keywords

free subalgebrasnon-associative algebrasLie algebrascommutative algebrasanticommutative algebrasspecial Jordan algebrasSchreier varietynon-commutative Poisson algebras

MSC classification

Primary: 17A50: Free algebras
Secondary: 17A01: General theory 17A30: Algebras satisfying other identities 08B20: Free algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 1 , February 2022 , pp. 89 - 101
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cohn, P.M., Free ideal rings and localization in general rings (Cambridge University Press, 2006).CrossRefGoogle Scholar
Kurosh, A., Non-associative free algebras and free products of algebras, Rec. Math. [Mat. Sbornik] N.S. 20 (1947), 239–262.Google Scholar
Lewin, J., On Schreier varieties of linear algebras, Trans. Amer. Math. Soc. 132 (1968), 553–562.CrossRefGoogle Scholar
Makar-Limanov, L. and Malcolmson, P., Free subalgebras of enveloping fields, Proc. Amer. Math. Soc. 111 (1991), 315–322.CrossRefGoogle Scholar
Mikhalev, A.A., Shpilrain, V. and Yu, J., Combinatorial methods: free groups, polynomials, and free algebras, CMS Books in Mathematics, Volume 19 (Springer-Verlag, New York, 2003).CrossRefGoogle Scholar
Reichstein, Z., On a question of Makar-Limanov, Proc. Amer. Math. Soc. 124 (1996), 17–19.CrossRefGoogle Scholar
Shirshov, A.I., Subalgebras of free Lie algebras, Mat. Sb. (N.S.) 33(75) (1953), 441–452.Google Scholar
Shirshov, A.I., Subalgebras of free commutative and free anticommutative algebras, Mat. Sb. (N.S.) 34(76) (1954), 81–88.Google Scholar
Smoktunowicz, A., Makar-Limanov's conjecture on free subalgebras, Adv. Math. 222 (2009), 2107–2116.CrossRefGoogle Scholar
Witt, E., Die Unterringe der freien Lieschen ring, Math. Z. 64 (1956), 195–216.CrossRefGoogle Scholar
Zhevlakov, K.A., Slin'ko, A.M., Shestakov, I.P. and Shirshov, A.I., Rings that are nearly associative, Pure and Applied Mathematics (Elsevier Science, 1982).Google Scholar