Geodesic
Relatively free associative Lie nilpotent algebras of rank~$3$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1937-1946.

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Let $\Phi$ be an arbitrary unital associative and commutative ring. The relatively free Lie nilpotent algebras with three generators over $\Phi$ are studied. The product theorem is proved: $T^{(n)}T^{(m)} \subseteq T^{(n + m-1)}$, where $T^{(n)}$ is a verbal ideal generated by the commutators of degree $n$. The identities of three variables that are satisfied in a free associative Lie nilpotent algebra of degree $n\geq 3$ are described. It is proved that the additive structure of the considered algebra is a free module over the ring $\Phi$.
Keywords: associative Lie nilpotent algebra, identity in three variables, torsion of a free ring. 
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     author = {S. V. Pchelintsev},
     title = {Relatively free associative {Lie} nilpotent algebras of rank~$3$},
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     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a33/}
}
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S. V. Pchelintsev. Relatively free associative Lie nilpotent algebras of rank~$3$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1937-1946. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a33/

[1] S. A. Jennings, “On rings whose associated Lie rings are nilpotent”, Bull. Amer. Math. Soc., 53:6 (1947), 593–597 | DOI | MR | Zbl

[2] V. N. Latyshev, “On choosing a base in one $T$-ideal”, Sib. Mat. Journal, 4:5 (1963), 1122–1127 | MR | Zbl

[3] V. N. Latyshev, “On the finite generation of $T$-ideal with an element $[x_1,x_2,x_3,x_4]$”, Sib. Mat. Journal, 6:6 (1965), 1432–1434 | MR | Zbl

[4] D. Krakowski, A. Regev, “The polynomial identities of the Grassman algebra”, Trans. Amer. Math. Soc., 181 (1973), 429–438 | MR | Zbl

[5] I. B. Volichenko, The $T$-ideal generated by the element $[x_1x_2 ,x_3 ,x_4]$, Preprint No 22, Inst. Math. Acad. Sci. Beloruss. SSR, 1978, 13 pp.

[6] A. S. Gordienko, “The codimension of the commutator of length $4$”, Usp. Mat. Nauk, 62:1 (2007), 191–192 | DOI | MR | Zbl

[7] A. V. Grishin, S. V. Pchelintsev, “On the centers of relatively free algebras with an identity of Lie nilpotency”, Sb. Math., 206:11 (2015), 1610–1627 | DOI | MR | Zbl

[8] A. V. Grishin, S. V. Pchelintsev, “Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6”, Math. Sb., 207:12 (2016), 54–72 | DOI | MR | Zbl

[9] R. Tyler, “On the lower central factors of a free associative rings”, Can. J. Math., 27:2 (1975), 434–438 | DOI | MR | Zbl

[10] V. M. Petrogradsky, “Codimension growth of strong Lie nilpotent associative algebras”, Comm. Algebra, 39:3 (2011), 918–928 | DOI | MR | Zbl

[11] N. Gupta, F. Levin, “On the Lie ideals of a ring”, J. Algebra, 81 (1983), 225–231 | DOI | MR | Zbl

[12] R. K. Sharma, J. B. Srivastava, “Lie ideals in group rings”, Journal of Pure and Applied Algebra, 63 (1990), 67–80 | DOI | MR | Zbl

[13] A. Bapat, D. Jordan, “Lower central series of free algebras in symmetric tensor categories”, J. Algebra, 373 (2013), 299–311 | DOI | MR | Zbl

[14] G. Deryabina, A. Krasilnikov, “On some products of commutators in an associative ring”, International Journal of Algebra and Computation, 29:2 (2019), 333–341 | DOI | MR | Zbl

[15] S. Bhupatiraju, P. Etingof, D. Jordan, W. Kuszmaul, J. Li, “Lower central series of a free associative algebra over the integers and finite fields”, J. Algebra, 372 (2012), 251–274 | DOI | MR | Zbl

[16] A. Krasilnikov, “The additive group of a Lie nilpotent associative ring”, J. Algebra, 392 (2013), 10–22 | DOI | MR | Zbl

[17] S. V. Pchelintsev, Relatively free associative algebras of ranks 2 and 3 with Lie nilpotency identity and systems of generators of some $T$-spaces, 20 pp., arXiv: (in Russian) 1801.07771 [math.RA]

[18] R. R. Dangovski, On the maximal containments of lower central series ideals, arXiv: 1509.08030 [math.RA]

[19] N. Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, 10, Interscience, New York–London, 1962 | MR | Zbl

[20] A. V. Grishin, L. M. Tsybula, A. A. Shokola, “On $T$-spaces and relations in relatively free, Lie nilpotent, associative algebras”, J. Math. Sci., 177:6 (2011), 868–877 | DOI | MR | Zbl

[21] A. I. Shirshov, “On free Lie rings”, Mat. Sat., 45:2 (1958), 113–122 | MR | Zbl

[22] S. V. Pchelintsev, “On the torsion of a free alternative ring”, Sib. Mat Journal, 32:6 (1991), 142–149 | MR | Zbl

[23] S. V. Pchelintsev, “Additive basis of relatively free associative algebra with the identity of Lie nilpotency of degree 5 and its applications”, Sib. Mat Journal (to appear) | MR