Definition 1

The tensor product of the right A-module E and the left A-module F, denoted by $E \otimes_A F$ or $E \otimes_A F$ (or simply $E \otimes F$ if no confusion is to be feared) is the quotient $\mathbf{Z}$-module $C/D$ (the quotient of the $\mathbf{Z}$-module C of formal linear combinations of elements of $E \times F$ with coefficients in $\mathbf{Z}$, by the submodule $D$ generated by the elements of one of the types (2)). *For $x \in E$ and $y \in F$, the element of $E \otimes_A F$ which is the canonical image of the element $(x, y)$ of $C = \mathbf{Z}^{(E \times F)}$ is denoted by $x \otimes y$ and called the tensor product of $x$ and $y$.

The mapping $(x, y) \mapsto x \otimes y$ of $E \times F$ into $E \otimes_A F$ is called canonical. It is a $\mathbf{Z}$-bilinear mapping which satisfies conditions (1).

We show that the tensor product $E \otimes_A F$ and the above canonical mapping form a solution of the universal mapping problem posed earlier. To be precise: