torsion abelian groups category

**xianghu21** · November 18th 2008, 08:25 PM

Let Tors be the category whose objects are torsion abelian groups; if $A$ and $B$ are torsion abelian groups, we define $Mor_{\text{Tors}}(A, B)$ to be the set of all (group) homomorphisms $\phi : A \rightarrow B$ . Prove that direct products exist in Tors; that is, show that given any indexed family $\{A_i\}_{i \in I}$ where $A_i$ is a torsion abelian group, there exists a torsion abelian group which serves as a direct product for this family in Tors.

**NonCommAlg** · November 18th 2008, 09:14 PM

Originally Posted by xianghu21

Let Tors be the category whose objects are torsion abelian groups; if $A$ and $B$ are torsion abelian groups, we define $Mor_{\text{Tors}}(A, B)$ to be the set of all (group) homomorphisms $\phi : A \rightarrow B$ . Prove that direct products exist in Tors; that is, show that given any indexed family $\{A_i\}_{i \in I}$ where $A_i$ is a torsion abelian group, there exists a torsion abelian group which serves as a direct product for this family in Tors.

let $B=\prod_{i \in I}A_i$ in the category of abelian groups, and let $T$ be the torsion subgroup of $B.$ now let $\pi_i: B \longrightarrow A_i$ be the projection map and $\tilde{\pi_i}$ be the restriction of $\pi_i$ to $T.$

then $\{T, \tilde{\pi}_i \}_{i \in I}$ is the product of $\{A_i \}_{i \in I}$ in the category $\text{Tors}.$

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