Varieties of Associative Algebras Satisfying Engel Identities

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.