Geodesic
On unsolvable $Q$-theories of ring varieties
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 18 (2018) no. 3, pp. 20-26

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Let $\mathcal{M}$ be any proper variety of associative rings. We prove that there exists an infinite set of varieties of associative rings containing $\mathcal{M}$ with unsolvable $Q$-theories. In particular, this result is a positive solution to the Mal'cev problem from the Kourovka Notebook on the existence of such varieties.
Keywords: quasivariety, variety, $Q$-theory, solvability, universal algebra, ring, Lee ring. 
@article{VNGU_2018_18_3_a1,
     author = {A. I. Budkin},
     title = {On unsolvable $Q$-theories of ring varieties},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {20--26},
     publisher = {mathdoc},
     volume = {18},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2018_18_3_a1/}
}
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A. I. Budkin. On unsolvable $Q$-theories of ring varieties. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 18 (2018) no. 3, pp. 20-26. http://geodesic.mathdoc.fr/item/VNGU_2018_18_3_a1/