## 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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### References:

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