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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
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
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
Linking options:
https://www.mathnet.ru/eng/sm646https://doi.org/10.1070/SM2002v193n04ABEH000646 https://www.mathnet.ru/eng/sm/v193/i4/p113
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Abstract page: | 454 | Russian version PDF: | 196 | English version PDF: | 17 | References: | 82 | First page: | 1 |
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