Definition:Tensor Product of Abelian Groups/Family/Definition 1
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Definition
Let $I$ be an indexing set.
Let $\family {G_i}_{i \mathop \in I}$ be a family of abelian groups.
Let $G = \ds \prod_{i \mathop \in I} G_i$ be their direct product.
Their tensor product is an ordered pair:
- $\struct {\ds \bigotimes_{i \mathop \in I} G_i, \theta}$
where:
- $\ds \bigotimes_{i \mathop \in I} G_i$ is an abelian group
- $\theta: G \to \ds \bigotimes_{i \mathop \in I} G_i$ is a multiadditive mapping such that, for every pair $\tuple {C, \omega}$ where:
- $C$ is an abelian group
- $\omega : G \to C$ is a multiadditive mapping
- there exists a unique group homomorphism $g : \ds \bigotimes_{i \mathop \in I} G_i \to C$ such that $\omega = g \circ \theta$.
- $\xymatrix{
G \ar[d]_\theta \ar[r]^\omega & C\\ \ds \bigotimes_{i \mathop \in I} G_i \ar@{.>}[ru]_g }$