Let $R$ be a ring with unit and consider a right $R$-module $M$ and a left $R$-module $N$. The tensor product $(M \otimes_R N, \otimes_R )$ of $M$ and $N$, is usually defined as the quotient of the free $\mathbb{Z}$-module $F$ generated by $M \times N$, and the submodule $H$ of $F$ generated by the elements that have the form of one of the following: $$\begin{align*}j(m + m', n) - j (m, n) - j (m', n), \\ j(m , n + n') - j (m, n) - j (m, n'), \\ j(m \lambda , n) - j(m, \lambda n) \end{align*}$$ where $m,m' \in M$, $n, n' \in N$, $\lambda \in R$, and $j : M \times N \to F$ is the inclusion. Finally, if $\psi : F \to F/H$ is the canonical $R$-morphism, we define $\otimes_R = \psi \circ j$. The book from which I got this definition is Blyth's Module Theory, which begins with stating the universal property of the tensor product but ultimately gets to this which I have stated. Then the book proceeds to say that $M \otimes_R N$ does not always have the structure of a left $R$-module.
My questions are, if in this construction we take $F$ as the free $R$-module generated by $M\times N$, would not it then have a structure of left $R$-module? And, why isn't it constructed in this manner?