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2021-10-20
AUTHORSM. Ramezan-Nassab, M. H. Bien, M. Akbari-Sehat
ABSTRACTLet R be an algebraic algebra over an infinite field and ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document} be an involution on R. We show that if the units of R, U(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}(R)$$\end{document}, satisfy a ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-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 U(FG)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}(FG)$$\end{document} satisfies a Laurent polynomial identity which is not satisfied by the units of the relative free algebra F[α,β:α2=β2=0]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F[\alpha , \beta :\alpha ^2=\beta ^2=0]$$\end{document}, then FG satisfies a polynomial identity. In this paper, we instead consider non-torsion groups G and provide some necessary conditions for U(FG)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}(FG)$$\end{document} to satisfy a Laurent polynomial identity. More... »
PAGES617-630
http://scigraph.springernature.com/pub.10.1007/s00013-021-01671-4
DOIhttp://dx.doi.org/10.1007/s00013-021-01671-4
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