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 Fundam. Prikl. Mat., 2007, Volume 13, Issue 1, Pages 135–159 (Mi fpm8)

Limit T-spaces

E. A. Kireeva

Moscow State Pedagogical University

Abstract: Let $F$ be a field of prime characteristic $p$ and let $\mathbf V_p$ be the variety of associative algebras over $F$ without unity defined by the identities $[[x,y],z]=0$ and $x^4=0$ if $p=2$ and by the identities $[[x,y],z]=0$ and $x^p=0$ if $p>2$ (here $[x,y]=xy-yx$). Let $A/V_p$ be the free algebra of countable rank of the variety $\mathbf V_p$ and let $S$ be the T-space in $A/V_p$ generated by $x_1^2x_2^2…x_k^2+V_2$, where $k\in\mathbb N$ if $p=2$ and by $x_1^{\alpha_1}x_2^{\alpha_2}[x_1,x_2]…x_{2k?1}^{\alpha_{2k-1}}x_{2k}^{\alpha_{2k}}[x_{2k?1},x_{2k}]+V_p$, where $k\in\mathbb N$ and $\alpha_1,…,\alpha_{2k}\in\{0,p-1\}$ if $p>2$. As is known, $S$ is not finitely generated as a T-space. In the present paper, we prove that $S$ is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative $F$-algebra without unity of countable rank.

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English version:
Journal of Mathematical Sciences (New York), 2008, 152:4, 540–557

Bibliographic databases:

UDC: 512.552

Citation: E. A. Kireeva, “Limit T-spaces”, Fundam. Prikl. Mat., 13:1 (2007), 135–159; J. Math. Sci., 152:4 (2008), 540–557

Citation in format AMSBIB
\Bibitem{Kir07} \by E.~A.~Kireeva \paper Limit T-spaces \jour Fundam. Prikl. Mat. \yr 2007 \vol 13 \issue 1 \pages 135--159 \mathnet{http://mi.mathnet.ru/fpm8} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2322963} \zmath{https://zbmath.org/?q=an:1161.16017} \transl \jour J. Math. Sci. \yr 2008 \vol 152 \issue 4 \pages 540--557 \crossref{https://doi.org/10.1007/s10958-008-9081-8} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-51749123542} 

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This publication is cited in the following articles:
1. A. V. Grishin, L. M. Tsybulya, “On the structure of a relatively free Grassmann algebra”, J. Math. Sci., 171:2 (2010), 149–212
2. Brandão A.P. (Jr.), Koshlukov P., Krasilnikov A., da Silva É.A., “The central polynomials for the Grassmann algebra”, Israel J. Math., 179:1 (2010), 127–144
3. Concalves D.J. Krasilnikov A. Sviridova I., “Limit T-Subalgebras in Free Associative Algebras”, J. Algebra, 412 (2014), 264–280
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