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Nilpotency in Bernstein algebras. (English) Zbl 0692.17013
A commutative nonassociative algebra A over a field K is called a Bernstein algebra if there is a nonzero algebra homomorphism \(\omega\) : \(A\to K\) such that \(x^ 2x^ 2=\omega (x)^ 2x^ 2\) for all \(x\in A\). Let (A,\(\omega)\) be a Bernstein algebra and denote by N the kernel of \(\omega\). It is known that if A is finitely generated then N is solvable but in general N is not nilpotent [the author and I. R. Hentzel, Arch. Math. 52, No.6, 539-543 (1989; Zbl 0643.17018)]. In this paper we prove that if A is finitely generated and \(A^ 2=A\) then N is nilpotent. This proves the truth of a conjecture posed by A. N. Grishkov [Dokl. Akad. Nauk SSSR 194, No.1, 27-30 (1987; Zbl 0643.17017)].
Reviewer: L.A.Peresi

MSC:
17D92 Genetic algebras
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References:
[1] A. N. Grishkov, On the genetic property of Bernstein algebras. Soviet Math. Dokl.35, 489-492 (1987). · Zbl 0643.17017
[2] I. R. Hentzel andL. A. Peresi, Semiprime Bernstein algebras. Arch. Math.52, 539-543 (1989). · Zbl 0643.17018 · doi:10.1007/BF01237566
[3] R. W. K.Odoni and A. E.Stratton, Structure of Bernstein algebras. Unpublished.
[4] A. W?rz-Busekros, Bernstein algebras. Arch. Math.48, 388-398 (1987). · Zbl 0597.17014 · doi:10.1007/BF01189631
[5] K. A.Zhevlakov et al, Rings that are nearly associative. New York-London 1982.
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