
Prime and homogeneous rings and algebras
E. I. Timoshenko^{} ^{} Novosibirsk State Technical University
Abstract:
Let $\mathcal M$ be a structure of a signature $\Sigma$. For any
ordered tuple $\overline{a}=(a_1,\ldots,a_n)$ of elements of
$\mathcal M$, $\mathrm{ tp}^{\mathcal M}(\overline{a})$ denotes the set
of formulas $\theta(x_1,\ldots,x_n)$ of a firstorder language over
$\Sigma$ with free variables $x_1,\ldots,x_n$ such that $\mathcal
M\models\theta(a_1,\ldots,a_n)$.
A structure $\mathcal M$ is said to
be strongly $\omega$homogeneous if, for any finite ordered tuples
$\overline{a}$ and $\overline{b}$ of elements of $\mathcal M$, the
coincidence of $\mathrm{ tp}^{\mathcal M}(\overline{a})$ and $\mathrm{
tp}^{\mathcal M}(\overline{b})$ implies that these tuples are mapped
into each other (componentwise) by some automorphism of the
structure $\mathcal M$. A structure $\mathcal M$ is said to be prime
in its theory if it is elementarily embedded in every structure of
the theory $\mathrm{ Th} (\mathcal M)$.
It is proved that the integral
group rings of finitely generated relatively free orderable groups
are prime in their theories, and that this property is shared by the
following finitely generated countable structures: free nilpotent
associative rings and algebras, free nilpotent rings and Lie
algebras. It is also shown that finitely generated nonAbelian free
nilpotent associative algebras and finitely generated nonAbelian
free nilpotent Lie algebras over uncountable fields are strongly
$\omega$homogeneous.
Keywords:
homogeneous structure, structure prime in its theory, relatively
free structure, orderable group, group ring, nilpotent algebra,
nilpotent ring, associative ring, Lie ring.
DOI:
https://doi.org/10.33048/alglog.2019.58.407
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UDC:
512.5 Received: 12.12.2018 Revised: 08.11.2019
Citation:
E. I. Timoshenko, “Prime and homogeneous rings and algebras”, Algebra Logika, 58:4 (2019), 512–527
Citation in format AMSBIB
\Bibitem{Tim19}
\by E.~I.~Timoshenko
\paper Prime and homogeneous rings and algebras
\jour Algebra Logika
\yr 2019
\vol 58
\issue 4
\pages 512527
\mathnet{http://mi.mathnet.ru/al913}
\crossref{https://doi.org/10.33048/alglog.2019.58.407}
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http://mi.mathnet.ru/eng/al913 http://mi.mathnet.ru/eng/al/v58/i4/p512
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