Abstract
Let R be an algebraic algebra over an infinite field and \(*\) be an involution on R. We show that if the units of R, \({\mathcal {U}}(R)\), satisfy a \(*\)-Laurent polynomial identity, then R satisfies a polynomial identity. Also, let G be a torsion group and F a field. As a generalization of Hartley’s Conjecture, in Broche et al. (Arch Math 111:353–367, 2018), it is shown that if \({\mathcal {U}}(FG)\) satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra \(F[\alpha , \beta :\alpha ^2=\beta ^2=0]\), then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for \({\mathcal {U}}(FG)\) to satisfy a Laurent polynomial identity.
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References
Amitsur, S.A.: Identities in rings with involutions. Israel J. Math. 7, 63–68 (1969)
Bien, M.H., Ramezan-Nassab, M.: Additive mappings and identities on unit groups of algebraic algebras. J. Algebra Appl., to appear. https://doi.org/10.1142/S021949882250116X
Broche, O., Goncalves, J.Z., del Rio, A.: Group algebras whose units satisfy a Laurent polynomial identity. Arch. Math. (Basel) 111, 353–367 (2018)
Dokuchaev, M.A., Goncalves, J.Z.: Identities on units of algebraic algebras. J. Algebra 250, 638–646 (2002)
Giambruno, A., Milies, C.P., Sehgal, S.K.: Star-group identities and groups of units. Arch. Math. (Basel) 95, 501–508 (2010)
Giambruno, A., Sehgal, S.K., Valenti, A.: Group identities on units of group algebras. J. Algebra 226, 488–504 (2000)
Giambruno, A., Sehgal, S.K., Valenti, A.: Group algebras whose units satisfy a group identity. Proc. Amer. Math. Soc. 125, 629–634 (1997)
Giambruno, A., Jespers, E., Valenti, A.: Group identities on units of rings. Arch. Math. (Basel) 63, 291–296 (1994)
Lee, G.T.: Group Identities on Units and Symmetric Units of Group Rings. Algebra and Applications, vol. 12. Springer, London (2010)
Liu, C.-H.: Some properties on rings with units satisfying a group identity. J. Algebra 232, 226–235 (2000)
Liu, C.-H.: Group identities on units of locally finite algebras and twisted group algebras. Comm. Algebra 28, 4783–4801 (2000)
Liu, C.-H.: Group algebras with units satisfying a group identity. Proc. Amer. Math. Soc. 127, 327–336 (1999)
Liu, C.-H., Passman, D.S.: Group algebras with units satisfying a group identity II. Proc. Amer. Math. Soc. 127, 337–341 (1999)
Milies, C.P., Sehgal, S.K.: An Introduction to Group Rings. Kluwer Academic Publishers, Dordrecht (2002)
Passman, D.S.: The Algebraic Structure of Group Rings. Wiley, New York (1977)
Acknowledgements
The authors are grateful to the referee for comments and suggestions that helped to improve the final version of this article. The second author is funded by Vietnam National University Ho Chi Minh City (VNUHCM) under Grant No. B2020-18-02.
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Ramezan-Nassab, M., Bien, M.H. & Akbari-Sehat, M. Algebras whose units satisfy a \(*\)-Laurent polynomial identity. Arch. Math. 117, 617–630 (2021). https://doi.org/10.1007/s00013-021-01671-4
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DOI: https://doi.org/10.1007/s00013-021-01671-4