Nilpotency of the alternator ideal of a finitely generated binary \((-1,1)\)-algebra. (Russian, English) Zbl 1059.17022

Sib. Mat. Zh. 45, No. 2, 427-451 (2004); translation in Sib. Math. J. 45, No. 2, 356-375 (2004).
The article under review is a continuation of the author’s article [Sb. Math. 193, No. 4, 585–607 (2002; Zbl 1035.17044)]. An algebra \(A\) is called a binary \((-1,1)\)-algebra if every two-generated subalgebra of \(A\) is a \((-1,1)\)-algebra. The author proves that the alternator ideal of a finitely generated binary \((-1,1)\)-algebra is nilpotent. As a corollary he obtains the following theorems: (1) a prime finitely generated binary \((-1,1)\)-algebra is alternative; (2) the Mikheev radical of a binary \((-1,1)\)-algebra coincides with the locally nilpotent radical; (3) a simple binary \((-1,1)\)-algebra is alternative; (4) the radical of a free finitely generated binary \((-1,1)\)-algebra is solvable; and (5) the radical of a finitely generated binary \((-1,1)\) PI-algebra is nilpotent.


17D20 \((\gamma, \delta)\)-rings, including \((1,-1)\)-rings
17D15 Right alternative rings


Zbl 1035.17044
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