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A fully invariant ideal of alternative algebras

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Abstract

Let ϕ be an associative commutative ring with 1, containing 1/6, and A be an alternative ϕ-algebra. Let D be an associator ideal of A and H a fully invariant ideal of A, generated by all elements of the form h(y, z, t, x, x)=[{[y, z], t, x}-, x]+[{[y, x], z, x}-, t], where [x, y]=xy−yx, {x, y, z}-=[[x, y], z]−[[x, z], y]+2[x,[y, z]]. Here we consider an ideal Q=H∩D and prove that Q4=0 in the algebra A. If A is unmixed, then HD=0, DH=0, and Q2=0 in particular. If A is a finitely generated unmixed algebra, then the ideal H lies in its associative center and Q=0. It follows that any finitely generated purely alternative algebra satisfies the identity h(y,z,t,x,x)=0. We also show that a fully invariant ideal H0 of the unmixed algebra A, generated by all elements of the form h(x, z, t, x, x), lies in its associative center and H0∩D=0. Consequently, every purely alternative algebra satisfies the identity h(x,z,t,x,x)=0.

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  1. V. T. Filippov
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Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 323–340, May–June, 1997.

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Filippov, V.T. A fully invariant ideal of alternative algebras. Algebr Logic 36, 193–203 (1997). https://doi.org/10.1007/BF02671617

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  • DOI: https://doi.org/10.1007/BF02671617

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