Geodesic
Artinian $\mathbf{M}$-complete, $\mathbf{M}$-reduced, and minimally $\mathbf{M}$-complete associative rings
Ural mathematical journal, Tome 10 (2024) no. 1, pp. 84-98

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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 
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Leonid M. Martynov; Tatiana V. Pavlova. Artinian $\mathbf{M}$-complete, $\mathbf{M}$-reduced, and minimally $\mathbf{M}$-complete associative rings. Ural mathematical journal, Tome 10 (2024) no. 1, pp. 84-98. http://geodesic.mathdoc.fr/item/UMJ_2024_10_1_a7/