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Sbornik: Mathematics, 2002, Volume 193, Issue 4, Pages 585–607
DOI: https://doi.org/10.1070/SM2002v193n04ABEH000646
(Mi sm646)
 

This article is cited in 7 scientific papers (total in 7 papers)

On central ideals of finitely generated binary $(-1,1)$-algebras

S. V. Pchelintsev

Moscow Pedagogical University, Moscow, Russian Federation
References:
Abstract: In 1975 the author proved that the centre of a free finitely generated $(-1,1)$-algebra contains a non-zero ideal of the whole algebra. Filippov proved that in a free alternative algebra of rank $\geqslant 4$ there exists a trivial ideal contained in the associative centre. Il'tyakov established that the associative nucleus of a free alternative algebra of rank 3 coincides with the ideal of identities of the Cayley–Dickson algebra.
In the present paper the above-mentioned theorem of the author is extended to free finitely generated binary $(-1,1)$-algebras.
Theorem. \textit{The centre of a free finitely generated binary $(-1,1)$-algebra of rank $\geqslant 3$ over a field of characteristic distinct from {\textrm2} and {\rm3} contains a non-zero ideal of the whole algebra.}
As a by-product, we shall prove that the $T$-ideal generated by the function $(z,x,(x,x,y))$ in a free binary $(-1,1)$-algebra of finite rank is soluble. We deduce from this that the basis rank of the variety of binary $(-1,1)$-algebras is infinite.
Received: 10.07.2001
Russian version:
Matematicheskii Sbornik, 2002, Volume 193, Number 4, Pages 113–134
DOI: https://doi.org/10.4213/sm646
Bibliographic databases:
UDC: 512.554.5
MSC: Primary 17D20; Secondary 17A50
Language: English
Original paper language: Russian
Citation: S. V. Pchelintsev, “On central ideals of finitely generated binary $(-1,1)$-algebras”, Mat. Sb., 193:4 (2002), 113–134; Sb. Math., 193:4 (2002), 585–607
Citation in format AMSBIB
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  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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