# 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:
• $\theta : G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$ is the composition of the canonical mapping $G \to \Z \sqbrk G$ with the quotient group epimorphism $\Z \sqbrk G \to \displaystyle \bigotimes_{i \mathop \in I} G_i$:
$G \hookrightarrow \Z \sqbrk G \twoheadrightarrow \displaystyle \bigotimes_{i \mathop \in I} G_i$

## Sources

• 1974: N. Bourbaki: Algebra I: Chapter $\text {II}$. Linear Algebra $\S 3$. Tensor Products. $9$. Tensor product of families of multimodules