Definition:Tensor Product of Abelian Groups/Family/Definition 2

Definition
Let $I$ be an indexing set.

Let $\family {G_i}_{i \mathop \in I}$ be a family of abelian groups.

Let $G = \displaystyle \prod_{i \mathop \in I} G_i$ be their direct product.

Their tensor product is the pair:
 * $\tuple {\displaystyle \bigotimes_{i \mathop \in I} G_i, \theta}$

where:
 * $\displaystyle \bigotimes_{i \in I} G_i$ is the quotient of the free abelian group $\Z \sqbrk G$ on $G$, by the subgroup generated by the elements of the form $\tuple {x + y, \family {z_i}_{i \mathop \ne j} } - \tuple {x, \family {z_i}_{i \mathop \ne j} } - \tuple {y, \family {z_i}_{i \mathop \ne j} }$
 * for $j \in I$, $x, y \in G_j$, $\family {z_i}_{i \mathop \ne j} \in \displaystyle \prod_{i \mathop \ne j} G_i$, where we denote $\tuple {x, \family {z_i}_{i \mathop \ne j} }$ for:
 * the family in $G$ whose $j$th term is $x$ and whose $i$th term is $z_i$, for $i \ne j$
 * its image under the canonical mapping $G \to \Z \sqbrk G$.
 * $\theta : G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$ is the composition of the canonical mapping $G \to \Z \sqbrk G$ with the quotient group epimorphism $\Z \sqbrk G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$:
 * $G \hookrightarrow \Z \sqbrk G \twoheadrightarrow \displaystyle \bigotimes_{i \mathop \in I} G_i$

Also see

 * Equivalence of Definitions of Tensor Product of Family of Abelian Groups