Definition:Tensor Product of Abelian Groups

Definition
Let $A$ and $B$ be abelian groups.

Definition 1: by universal property
Their tensor product is a pair $(A \otimes B, \theta)$ where: such that, for every pair $(C, \omega)$ where: there exists a unique group homomorphism $g : A \otimes B \to C$ with $\omega = g \circ \theta$.
 * $A \otimes B$ is an abelian group
 * $\theta : A \times B \to A \otimes B$ is a biadditive mapping
 * $C$ is an abelian group
 * $\omega : A \times B \to C$ is a biadditive mapping

Definition 2: construction
Their tensor product is the pair $(A \otimes B, \theta)$ where:
 * $A \otimes B$ is the quotient of the free abelian group $\Z^{(A \times B)}$ on the cartesian product $A \times B$ by the subgroup generated by the elements of the form:
 * $(a_1 + a_2, b) - (a_1, b) - (a_2, b)$
 * $(a, b_1 + b_2) - (a, b_1) - (a, b_2)$
 * for $a, a_1, a_2 \in A$, $b, b_1, b_2 \in B$, where we denote $(a, b)$ for its image under the canonical mapping $A \times B \to \Z^{(A\times B)}$.
 * $\theta : A \times B \to A \otimes B$ is the composition of the canonical mapping $A \times B \to \Z^{(A\times B)}$ with the quotient group epimorphism $\Z^{(A\times B)} \to A \otimes B$.

Also see

 * Definition:Tensor Product of Modules as Abelian Group
 * Definition:Tensor Product of Modules