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Central ideals of a free finitely generated alternative algebra. (English. Russian original) Zbl 0635.17014
Algebra Logic 25, 296-311 (1986); translation from Algebra Logika 25, No. 4, 470-491 (1986).
Let \(A_ k\) be a free alternative algebra on k free generators over an associative-commutative ring \(\Phi\) containing 1/6. Consider the following function (Filippov’s function): \[ g(y,z,t,v,x)=([[y,z],t,x],x,v)+([[y,x],z,x],t,v), \] where \([u,v,w]=2(u,v,w)+[u,[v,w]]\), and next define \[ \omega_ k(x_ 1,...,x_{k+1})=[... [[g(x_ 1,...,x_ 5),x_ 6],x_ 7],...,x_{k+1}\}. \] It is proved that all the values of the function \(\omega_ k\) generate a T-ideal \(W(A_ k)\) in the algebra \(A_ k\) (k\(\geq 5)\), which is trivial (i.e. \(W(A_ k)^ 2=0)\) and lies in the center \(Z(A_ k)\); at the same time \(W(A_{k+1})\not\subseteq Z(A_{k+1})\).
Reviewer: I.Shestakov
MSC:
17D05 Alternative rings
17D10 Mal’tsev rings and algebras
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References:
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