Artinian $\mathbf{M}$-complete, $\mathbf{M}$-reduced, and minimally $\mathbf{M}$-complete associative rings

In 1996, the first author defined analogs of the concepts of complete (divisible), reduced, and periodic abelian groups, well-known in the theory of abelian groups, for arbitrary varieties of algebras. In 2021, the first author proposed a modification of the concepts of completeness and reducibility, which is more natural in the case of associative rings. The paper studies the modification of these concepts for associative rings. Artinian $\mathbf{M}$-complete, $\mathbf{M}$-reduced rings, and minimally $\mathbf{M}$-complete associative nilpotent rings, simple rings with unity, and finite rings are characterized.