Geodesic
Almost Lie nilpotent varieties of associative rings
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 901-909.

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A variety of associative rings is called Lie nilpotent if it satisfies the identity $[\dots[[x_1, x_2],\dots,x_n]=0$ for some positive integer $n$, where $[x, y]=xy-yx$. We study almost Lie nilpotent varieties, i.e., minimal elements in the set of all varieties that are not Lie nilpotent. We reduce the case of rings to the case of algebras over a finite prime field by proving that every almost Lie nilpotent variety of rings satisfies the identity $px=0$ for some prime integer $p$. We also show that for every finite base field $F$ it is sufficient to study all prime almost Lie nilpotent varieties algebras over any infinite extension of $F$ to find all such varieties of $F$-algebras. The nonprime almost Lie nilpotent varieties of algebras over positive characteristic fields, both infinite and finite, were described by the author in an earlier paper.
Keywords: Variety of associative algebras, identities of the associated Lie algebra, Lie nilpotency, Engel property, prime variety. 
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     author = {O. B. Finogenova},
     title = {Almost {Lie} nilpotent varieties of associative rings},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {901--909},
     publisher = {mathdoc},
     volume = {12},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2015_12_a28/}
}
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O. B. Finogenova. Almost Lie nilpotent varieties of associative rings. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 12 (2015), pp. 901-909. http://geodesic.mathdoc.fr/item/SEMR_2015_12_a28/

[1] Algebra i Logika, 43:4 (2004), 482–505 | DOI | MR | Zbl

[2] Algebra i Logika, 52 (2013), 731–768 | DOI | MR | Zbl

[3] O. B. Finogenova, “Almost Lie nilpotent non-prime varieties of associative algebras”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 21, no. 4, 2015, 282–291 (in Russian)

[4] Algebra i Logika, 19 (1980), 255–283 | DOI | MR | Zbl

[5] A. R. Kemer, “Multilinear components of the prime subvarieties of the variety $VarM_2(F)$”, Algebras and Representation Theory, 4 (2001), 87–104 | DOI | MR | Zbl

[6] Algebra i Logika, 15 (1976), 579–584 | DOI | MR | Zbl | Zbl

[7] Yu. P. Razmyslov, “Identities of algebras and their representations”, Translations of Math. Monographs, 138, American Mathematical Society, Providence, RI, 1994 ; Algebra, Proc. Int. Conf. Memory A. I. Mal'cev (Novosibirsk/USSR, 1989), v. 2, Contemp. Math., 131, 1992, 173–192 | MR | Zbl | DOI | Zbl

[8] Sib. Mat. Zh., 51 (2010), 890–903 | DOI | MR | Zbl

[9] Sib. Mat. Zh., 29 (1988), 112–117 | DOI | MR | Zbl