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A useful category for mixed Abelian groups. (English) Zbl 0919.20035
The author introduces a new category denoted by $$\mathcal A$$ whose objects are all abelian groups and whose morphisms are $${\mathcal A}(G,H)=\mathbf{Ab}(G,H)/{\mathbf J}(G,H)$$, where $${\mathbf J}(G,H)=\{f\colon G\to H\mid T(G)\subseteq\text{Ker}(f)\}$$ for $$G,H\in\mathbf{Ob}({\mathcal A})$$. It is shown that $$\mathcal A$$ is an additive category which has cokernels, products and infinite coproducts but which does not have kernels. Moreover two abelian groups $$G$$ and $$H$$ are $$\mathcal A$$-isomorphic if and only if there are two torsionfree abelian groups $$U$$ and $$V$$ such that $$G\oplus U\cong H\oplus V$$. Thus all torsionfree groups are $$\mathcal A$$-isomorphic and the category $$\mathcal A$$ ignores torsion-freeness. Furthermore, there is a ring embedding which sends the $$\mathcal A$$-endomorphism ring $$\text{End}_{{\mathcal A}}(G)$$ of some abelian group $$G$$ to $$\text{End}(T(G))$$, where $$T(G)$$ is the torsion part of $$G$$. This embedding is a ring isomorphism if $$G$$ splits.
In the second part the author considers the relationships between the category $$\mathcal A$$ and other useful categories. A natural functor $$\mathbf I$$ from $$\mathbf{To}$$, the full subcategory of $$\mathbf{Ab}$$ which consists of all the torsion abelian groups into $$\mathcal A$$ is constructed and it is shown that $$\mathbf I$$ has a right adjoint. The properties of $$\mathbf I$$ are determined and a necessary and sufficient condition for $$\mathbf I$$ to be an equivalence is given. Finally the author considers the category $$\mathbf{Walk}$$ which consists of all abelian groups as objects and the morphisms are $$\mathbf{Walk}(G,H)=\mathbf{Ab}(G,H)/{\mathbf I}(G,H)$$, where $${\mathbf I}(G,H)=\{f\in\mathbf{Ab}(G,H)\mid\text{Im}(f)\subseteq T(H)\}$$. It is proved that if $$G\cong H$$ in $$\mathbf{Walk}$$ and $$G\cong H$$ in $$\mathcal A$$ then $$T(G)\cong T(H)$$ and $$G/T(G)\cong H/T(H)$$. Thus the class of splitting mixed groups is characterized. Some open problems are stated at the end.
##### MSC:
 20K21 Mixed groups 18E05 Preadditive, additive categories 20K40 Homological and categorical methods for abelian groups
##### Keywords:
mixed Abelian groups; category Walk; additive categories
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