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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
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