Motivation behind tensor product construction

Question

At the beginning of Dummit and Foote section 10.4, in order to extend a $R$ module $N$ to an $S$ module, where $R$ is a subring of $S$, we start with the most general setting where this is possible: a free $\mathbb{Z}$ module on the product $S \times N$. Then, we take the quotient of this group by subgroup generated by elements of the form $$ (s_1+s_2, n) - (s_1,n)-(s_2,n),\\ (s,n_1+n_2)-(s,n_1)-(s,n_2),\\ (sr,n) - (s,rn), $$ where $s,s_1,s_2 \in S, r \in R, n,n_1,n_2 \in N$. My question is why in the third kind of elements we are restricting ourselves to elements of the form $sr$ with $s \in S$ and $r \in R$ only. For $s_1, s_2 \in S$ and $n \in N$, we also have (after the construction has completed) that $(s_1s_2)n = s_1(s_2n)$. Why not consider these elements in the quotienting process?