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Mat. Sb., 2002, Volume 193, Number 4, Pages 113–134 (Mi msb646)  

This article is cited in 5 scientific papers (total in 5 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.

DOI: https://doi.org/10.4213/sm646

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English version:
Sbornik: Mathematics, 2002, 193:4, 585–607

Bibliographic databases:

UDC: 512.554.5
MSC: Primary 17D20; Secondary 17A50
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

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. V. Pchelintsev, “Nilpotency of the alternator ideal of a finitely generated binary $(-1,1)$-algebra”, Siberian Math. J., 45:2 (2004), 356–375  mathnet  crossref  mathscinet  zmath  isi  elib
    2. S. V. Pchelintsev, “Irreducible binary $(-1,1)$-bimodules over simple finite-dimensional algebras”, Siberian Math. J., 47:5 (2006), 934–939  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    3. S. V. Pchelintsev, “Isotopes of prime $(-1,1)$- and Jordan algebras”, Algebra and Logic, 49:3 (2010), 262–288  mathnet  crossref  mathscinet  zmath  elib  elib
    4. S. V. Pchelintsev, “Prime algebras connected with monsters”, Siberian Math. J., 59:2 (2018), 341–356  mathnet  crossref  crossref  isi  elib
    5. L. R. Borisova, S. V. Pchelintsev, “O strukture alternativnykh bimodulei nad poluprostymi artinovymi algebrami”, Izv. vuzov. Matem., 2020, no. 8, 3–10  mathnet  crossref
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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