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A category of matrices representing two categories of Abelian groups. (English. Russian original) Zbl 1166.20051

J. Math. Sci., New York 154, No. 3, 430-445 (2008); translation from Fundam. Prikl. Mat. 13, No. 3, 223-244 (2007).
The author constructs a commutative triangle of categories and functors. One vertex is the category \(\mathcal{RM}\) of equivalence classes of reduced matrices over the ring of universal numbers, with multiplication as morphisms. Another is the category \(\mathcal{QD}\) of (mixed) quotient divisible Abelian groups with quasi-homomorphisms as morphisms. The third is the category \(\mathcal{QTF}\) of finite rank torsion-free Abelian groups with quasi-homomorphisms as morphisms. There is an equivalence from \(\mathcal{RM}\) to \(\mathcal{QD}\) and a duality from \(\mathcal{RM}\) to \(\mathcal{QTF}\). The diagram is closed by the induced duality from \(\mathcal{QD}\) to \(\mathcal{QTF}\).
The author derives many applications and establishes connections with Pontryagin duality and the Kurosh-Malcev-Derry classification of \(\mathcal{QTF}\).

MSC:

20K40 Homological and categorical methods for abelian groups
20K21 Mixed groups
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