On central ideals of finitely generated binary $(-1,1)$-algebras

S. V. Pchelintsev

Sbornik. Mathematics, Tome 193 (2002) no. 4, pp. 585-607

Résumé

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.