Abstract:
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.
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