The solved theories of the ring varieties are investigated. The existence of the finite-based varieties on the metabelian and commutative rings with non-solved equational theory has been proved, thereby the known A.I. Maltsev's problem has been solved. The chain of omega type consisting of finite-based varieties in the non-associative rings has been designed, in this case, it is such that any two near-standing varieties have the equational theories one of which is solved but other is non-solved. The solveability boundary types of the varieties in the above-associative-cummutative rings have been described. The critical theories on the variety of the nilpotent rings have been found. The results can be used in the further investigations on the algorithmic algebra problems, at lecturing of the special courses and writting of the hand-books in the Moscow, Omsk, Novosibirsk and Urals state universities and in the Institute of Mathematics and Mechanics, Urals Division of the Russian Academy of SciencesAvailable from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio
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