Existence of a tensor product

Let \(R\) be a ring and \(M,N\) \(R\)-module Then there exists a tensor product \(M \otimes_R N\).

Let \(F(M,N) = R^{M \times N}\) be the free module on \(M \times N\) as a set. Consider the set theoretic map

\begin{align*} \iota: M \times N \twoheadrightarrow& F(M,N) \\ (a,b) \mapsto& (1_{(a,b)},0) \end{align*}

Let \(V\) be generated by the subset

\begin{align*} \{\iota(m + m',n) - \iota(m,n) - \iota(m',n), \iota(m,n + n') - \iota(m,n) - \iota(m,n'), \iota(rm,n) - r\iota(m,n) \iota(m,rn) - r\iota(m,n)\} \end{align*}

By construction, the composite

\begin{align*} M \times N \underrightarrow{\iota} F(M,N) -> F(M,N) / V = M \otimes_{R} N \end{align*}

is a bilinear map.