Tensor Product of Abelian Groups

Let $M,N$ be a pair of Abelian Groups. The tensor product $M\otimes N$ is defined as the quotient of the Free Abelian Group on $M\times N$, quotiented by the following relations:

• $(a_{1},b)+(a_{2},b)\sim(a_{1}+a_{2},b)$
• $(a,b_{1})+(a,b_{2})\sim(a,b_{1}+b_{2})$

The tensor product $M\otimes N$ enjoys a universal property: maps $M\otimes N\to A$ are equivalent to Bilinear Maps $M\times N\to A$.