## 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.

### MSC:

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

Zbl 1035.17044
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