I have troubles in understanding the construction of the extension of scalars as done in Dummit & Foote Abstract Algebra, Chapter 10.4. The situation is the following: we have a subring $R$ of a ring $S$ with $1_R=1_S$ and we have a left $R$-module $N$. Then we want to construct an $S$-module in which to embed $N$ as an $R$-submodule, and the best choice seems to be the tensor product $S \otimes_R N$. Now I have some questions:
1) why is it the best choice? Is it because of the universal property? what is the meaning of the universal property?
2) if the tensor product is a quotient by the subgroup of the relations $H$ of the free $\mathbb{Z}$-module on the set $S\times N$, why can we write its elements simply as finite sums of simple tensors, forgetting the fact that there can also be negative multiples of certain simple tensors?
3) I don't understand the left $S$-module structure defined on $S \otimes_R N$ by $s(\sum_{i=1}^p s_i\otimes n_i) = \sum_{i=1}^p (ss_i)\otimes n_i$. In particular I don't understand why this action should be well-defined. What I've understood is that if $\sum s_i\otimes n_i=\sum s'_j\otimes n'_j$ then $\sum (s_i, n_i)-\sum (s'_j, n'_j) \in H$. But now I don't understand why multiplying the first entries of the couples by $s$ corresponds to multiplying by $s$ the first entries of the generators of $H$.
Thanks in advance for your help!