Geodesic
Finiteness of the standard basis of a $T$-ideal containing Lie nilpotency of index $4$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 75-85 
V. N. Latyshev. Finiteness of the standard basis of a $T$-ideal containing Lie nilpotency of index $4$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 75-85. http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a4/
@article{FPM_2012_17_5_a4,
     author = {V. N. Latyshev},
     title = {Finiteness of the standard basis of a~$T$-ideal containing {Lie} nilpotency of index~$4$},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {75--85},
     year = {2012},
     volume = {17},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a4/}
}
TY  - JOUR
AU  - V. N. Latyshev
TI  - Finiteness of the standard basis of a $T$-ideal containing Lie nilpotency of index $4$
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2012
SP  - 75
EP  - 85
VL  - 17
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a4/
LA  - ru
ID  - FPM_2012_17_5_a4
ER  - 
%0 Journal Article
%A V. N. Latyshev
%T Finiteness of the standard basis of a $T$-ideal containing Lie nilpotency of index $4$
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2012
%P 75-85
%V 17
%N 5
%U http://geodesic.mathdoc.fr/item/FPM_2012_17_5_a4/
%G ru
%F FPM_2012_17_5_a4

Voir la notice de l'article provenant de la source Math-Net.Ru

Recently the author presented a notion of standard basis for a $T$-ideal of the free associative algebra over a field of zero characteristic. It was proved that it is finite if the $T$-ideal contains either Lie nilpotency of index $3$ or a multilinear product of commutators of degree $2$. Here we prove the finiteness of the reduced standard basis of any $T$-ideal containing Lie nilpotency of index $4$.

[1] Volichenko I. B., $T$-ideal, porozhdënnyi elementom $[x_1,x_2,x_3,x_4]$, Preprint, In-t mat. AN BSSR, Minsk, 1978

[2] Kemer A. R., “Konechnye bazisy tozhdestv assotsiativnykh algebr”, Algebra i logika, 26 (1987), 597–641 | MR | Zbl

[3] Latyshev V. N., “O konechnoi porozhdënnosti $T$-ideala s elementom $[x_1,x_2,x_3,x_4]$”, Sib. mat. zhurn., 6:6 (1965), 1432–1434 | Zbl

[4] Latyshev V. N., “Kombinatornye porozhdayuschie polilineinykh polinomialnykh tozhdestv”, Fundament. i prikl. mat., 12:2 (2006), 101–110 | MR | Zbl

[5] Higman G., “Ordering by divisibility in abstract algebras”, Proc. Lond. Math. Soc., 2 (1952), 326–336 | DOI | MR | Zbl

[6] Specht W., “Gesetze in Ringen”, Math. Z., 52 (1950), 557–589 | DOI | MR | Zbl