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

Definition
Let $I$ be an indexing set.

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

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

Their tensor product is the pair $\left( \displaystyle \bigotimes_{i \in I} G_i, \theta \right)$ where:
 * $\displaystyle \bigotimes_{i \in I} G_i$ is the quotient of the free abelian group $\Z\left[ G \right]$ on $G$, by the subgroup generated by the elements of the form $\left( x + y, (z_i)_{i \neq j} \right) - \left( x, (z_i)_{i \neq j} \right) - \left( y, (z_i)_{i \neq j} \right)$
 * for $j\in I$, $x, y \in G_j$, $(z_i)_{i \neq j} \in \displaystyle \prod_{i \neq j} G_i$, where we denote $(x, (z_i)_{i \neq j})$ for:
 * the family in $G$ whose $j$th term is $x$ and whose $i$th term is $z_i$, for $i \neq j$
 * its image under the canonical mapping $G \to \Z[G]$.
 * $\theta : G \to \bigotimes_{i \in I} G_i$ is the composition of the canonical mapping $G \to \Z\left[ G \right]$ with the quotient group epimorphism $\Z\left[ G \right] \to \displaystyle\bigotimes_{i \in I} G_i$:
 * $G \hookrightarrow \Z\left[ G \right] \twoheadrightarrow \displaystyle\bigotimes_{i \in I} G_i$

Also see

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