ARTINIAN \(\mathbf{M}\)-COMPLETE, \(\mathbf{M}\)-REDUCED, AND MINIMALLY \(\mathbf{M}\)-COMPLETE ASSOCIATIVE RINGS

Abstract

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.

Keywords

• associative ring, artinian ring, finite ring, complete ring, reduced ring