Geodesic
On identities of free finitely generated alternative algebras over a~field of characteristic~3
Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1365-1380

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In 1981 Filippov solved in the affirmative Shestakov's problem on the strictness of the inclusions in the chains of varieties generated by free alternative and Mal'cev algebras of finite rank over a field of characteristic distinct from 2 and 3. In the present paper an analogous result is proved for alternative algebras over a field of characteristic 3. The proof is based on the construction of three families of identities that hold on the algebras of the corresponding rank. A disproof of the identities on algebras of larger rank is carried out with the help of a prime commutative alternative algebra. It is also proved that in varieties of alternative algebras of finite basis rank over a field of characteristic 3 every soluble algebra is nilpotent. 
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     author = {S. V. Pchelintsev},
     title = {On identities of free finitely generated alternative algebras over a~field of characteristic~3},
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S. V. Pchelintsev. On identities of free finitely generated alternative algebras over a~field of characteristic~3. Sbornik. Mathematics, Tome 192 (2001) no. 9, pp. 1365-1380. http://geodesic.mathdoc.fr/item/SM_2001_192_9_a4/