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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_2001_192_9_a4,
     author = {S. V. Pchelintsev},
     title = {On identities of free finitely generated alternative algebras over a~field of characteristic~3},
     journal = {Sbornik. Mathematics},
     pages = {1365--1380},
     year = {2001},
     volume = {192},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_9_a4/}
}
TY  - JOUR
AU  - S. V. Pchelintsev
TI  - On identities of free finitely generated alternative algebras over a field of characteristic 3
JO  - Sbornik. Mathematics
PY  - 2001
SP  - 1365
EP  - 1380
VL  - 192
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2001_192_9_a4/
LA  - en
ID  - SM_2001_192_9_a4
ER  - 
%0 Journal Article
%A S. V. Pchelintsev
%T On identities of free finitely generated alternative algebras over a field of characteristic 3
%J Sbornik. Mathematics
%D 2001
%P 1365-1380
%V 192
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2001_192_9_a4/
%G en
%F SM_2001_192_9_a4
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/

[1] Shestakov I. P., “Ob odnoi probleme Shirshova”, Algebra i logika, 16:2 (1977), 227–246 | MR | Zbl

[2] Filippov V. T., “O tsepochkakh mnogoobrazii, porozhdennykh svobodnymi maltsevskimi i alternativnymi algebrami”, Dokl. AN SSSR, 260:5 (1981), 1082–1085 | MR | Zbl

[3] Zhevlakov K. A., Slinko A. M., Shestakov I. P., Shirshov A. I., Koltsa, blizkie k assotsiativnym, Nauka, M., 1978 | MR | Zbl

[4] Pchelintsev S. V., “O kruchenii svobodnogo alternativnogo koltsa”, Sib. matem. zhurn., 32:6 (1991), 142–149 | MR

[5] Pchelintsev S. V., “O nilpotentnykh elementakh i nil-radikalakh alternativnykh algebr”, Algebra i logika, 24:6 (1985), 674–695 | MR | Zbl

[6] Pchelintsev S. V., “Teorema o vysote dlya alternativnykh algebr”, Matem. sb., 124:4 (1984), 557–567 | MR | Zbl

[7] Pchelintsev S. V., “Razreshimost i nilpotentnost alternativnykh algebr i algebr tipa $(-1,1)$”, Gruppy i drugie algebraicheskie sistemy s usloviyami konechnosti, Nauka, Novosibirsk, 1984, 81–101 | MR

[8] Shestakov I. P., “Alternativnye algebry s tozhdestvom $[x,y]^m=0$”, Algebra i logika, 20:5 (1981), 575–596 | MR | Zbl

[9] Dorofeev G. V., “O lokalno nilpotentnom radikale neassotsiativnykh kolets”, Algebra i logika, 10:4 (1971), 355–364 | MR | Zbl