On central ideals of finitely generated binary (-1,1)-algebras
- Moscow Pedagogical University, Moscow (Russian Federation)
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 (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.
- OSTI ID:
- 21205669
- Journal Information:
- Sbornik. Mathematics, Vol. 193, Issue 4; Other Information: DOI: 10.1070/SM2002v193n04ABEH000646; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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