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 ≥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. The centre of a free finitely generated binary (–1,1)-algebra of rank ≥3 over a field of characteristic distinct from 2 and 3 contains a non-zero ideal of the whole algebra. As a by-product, we shall prove that the T-ideal generated by the function 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.
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