Suppose $R$ and $S$ are rings such that $S$ contains $R$ and they both have the same identity. Let $N$ be a (left) $R$-module. I was trying to construct the tensor product $S\otimes_R N$ and also trying to show that this tensor product has an $S$-module structure.
The way I am doing it is the usual way: take the free abelian group generated by the set $S\times N$ and quotient it by a certain subgroup $Y$. The coset to which an element $(s,n)$ belongs is denoted by $s\otimes n$, and quotienting gives the following three relations: $$(s_1 + s_2)\otimes n = s_1\otimes n + s_2\otimes n\\ s\otimes (n_1 + n_2) = s\otimes n_1 + s\otimes n_2\\ (sr)\otimes n = s\otimes(rn) $$ and I don't have any issues till here.
Now, the action of $S$ on this quotient is defined as $$s\left(\sum_{i}s_i\otimes n_i\right) = \sum_{i}(ss_i)\otimes n_i$$
and the first job is to show that this is well defined; i.e no matter what representatives are chosen for the tensors, this action produces the same result. But I am having a difficulty with this.
Suppose $s_1\otimes n_1 = s_2\otimes n_2$, which means that $(s_1,n_1) - (s_2,n_2)\in Y$, and it is enough to show that $(ss_1,n_1) - (ss_2,n_2)\in Y$ as well. But I am not able to show this.
The source I am referring to just says this: $Y$ is generated by all elements of the form $(s_1 + s_2,n_1) - (s_1,n_1) - (s_2,n_1)$, $(s_1,n_1 + n_2) - (s_1,n_1) - (s_1,n_2)$ and $(s_1r,n_1) - (s_1,rn_1)$. If we multiply the first coordinates of any of these elements by some $s\in S$, then they still remain in $Y$. But I don't see how this is enough: in particular, if $(s_1,n_1) - (s_2,n_2)\in Y$, then $(s_1,n_1) - (s_2,n_2)$ can be written as the sum of elements of the given form, but how can we just multiply the first coordinates by $s$ and say we are done? For instance, if say $(s_1,n_1) - (s_2,n_2) = (s'r',n') - (s',r'n')$, then how can we say that $(ss_1,n_1) - (ss_2,n_2) = (ss'r',n') - (ss',r'n')$? Since $X$ is a free group, how can such a relation make sense?
I tried to search about this a lot, but most sources skip this step. Can you guys help me with this?