Geodesic
Varieties of Associative Algebras Satisfying Engel Identities
Algebra i logika, Tome 43 (2004) no. 4, pp. 482-505

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A variety of associative algebras (rings) is said to be Engel if it satisfies an identity of the form $[\ldots[[x,y],y],\ldots,y]=0$. On the Zorn lemma, every non-Engel variety contains some just non-Engel variety, that is, a minimal (w.r.t. inclusion) element in the set of all non-Engel varieties. A list of such varieties for algebras over a field of characteristic 0 was made up by Yu. N. Mal'tsev. Here, we present a complete description of just non-Engel varieties both for the case of algebras over a field of positive characteristic and for the case of rings. This gives the answer to Question 3.53 in the Dniester Notebook.
Keywords: Engel identity, just non-Engel variety, variety of associative rings, associative algebra over a field. 
@article{AL_2004_43_4_a5,
     author = {O. B. Finogenova},
     title = {Varieties of {Associative} {Algebras} {Satisfying} {Engel} {Identities}},
     journal = {Algebra i logika},
     pages = {482--505},
     publisher = {mathdoc},
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     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2004_43_4_a5/}
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O. B. Finogenova. Varieties of Associative Algebras Satisfying Engel Identities. Algebra i logika, Tome 43 (2004) no. 4, pp. 482-505. http://geodesic.mathdoc.fr/item/AL_2004_43_4_a5/