Tensor product (of modules)
Definition
Let $A$ be a commutative ring, and let $M$, $N$, and $K$ be $A$-modules.
Definition. Then tensor product $M\otimes_A N$ is an $A$-module satisfying the following universal property:
$$
\begin{array}
zM\times N & \rightarrow & M\otimes_A N \
& \llap{{}_B}{\searrow} & \rlap{\ \ {}^\exists}{\downarrow} \
& & k
\end{array}
$$
where $B$ is a bilinear map.
That is two say, there is a bijection $$ {M\times N\overset{B}{\to}K}\ \leftrightarrow \ \text{Hom}_A(M\otimes_A N, K). $$
Remark. This coincides with the notion of Tensor product of vector spaces. Notably, it suffices to consider $K=\mathbb{F}$.
Construction
Consider the free module $A^\alpha$, whose elements are finite linear combinations $$\sum_i a_i [m,n]$$ where $m\in M$ and $n\in N$, i.e. whose basis elements are those of the form $[m,n]$. Let $R$ be the submodule generated by elements of the form
\begin{gather}
[m_1+m_2,n]-[m_1,n]-[m_2,n],\
[m,an]-[ma,n],\
[m,an]-a[m,n].
\end{gather}
Then $$ A/R\cong M\otimes_A N. $$