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 $\struct {A \otimes B, \theta}$ where:
 * $A \otimes B$ is an abelian group
 * $\theta : A \times B \to A \otimes B$ is a biadditive mapping such that, for every ordered pair $\struct {C, \omega}$ where:
 * $C$ is an abelian group
 * $\omega : A \times B \to C$ is a biadditive mapping

there exists a unique group homomorphism $g : A \otimes B \to C$ such that $\omega = g \circ \theta$.

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

Also see

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