How is the action of scalar ring $S$ on $M\otimes_R N$ well-defined? Let $\sum^k m_i \otimes n_i  = \sum^l m_j' \otimes n_j'$ be two representations of the same tensor in the abelian group $M \otimes_R N$, where $M$ is an $(S,R)$-bimodule and  and a left $R$-module.  How is the action of $S$ on $M\otimes_R N$, defined by
$$
s(\sum^k m_i \otimes n_i) = \sum^k (s m_i)\otimes n_i
$$
well-defined?  I'm having trouble proving that.  The purpose is to show that $M \otimes_R N$ has a natural left $S$-module structure.
 A: Hint: You could use the universal property of the tensor product.
Let's look at $\ell_s:M\times N\to M\otimes_R N$ where $\ell_s(m,n):=(sm)\otimes n$. It's definitely well-defined since the module operation on $M$ is well-defined. What other properties does this map have?
Of course after all is said and done we're going to relabel $\ell_s(m,n)$ as $s(m,n)$. I just want you to concentrate on left multiplication by $s$ as a function.
A: It is sufficent to show that $s( m \otimes n) = (s m)\otimes n$. As you know, we mean from $m \otimes n$ is $(m,n)+L$ where $L$ is $\rm{Im}\otimes$.
Now, for showing  $s( m \otimes n) = (s m)\otimes n$:
$s( m \otimes n) =s(m,n)+L=(sm,n)+L=(s m)\otimes n$
A: Don't use elements, because then you will miss the obvious and elegant proof: $M$ is an $(S,R)$-bimodule, this means that we have an right $R$-linear map $S \otimes_R M \to M$ satisfying the usual axioms. The tensor product is associative (this is what happens in rschwieb's answer, but why proving this once again in a special case?) and functorial, hence we get an additive map
$$S \otimes_R (M \otimes_R N) \cong (S \otimes_R M) \otimes_R N \longrightarrow M \otimes_R N.$$
The left $S$-module axioms can be checked using diagram cases. In fact, the whole construction works in an arbitrary cocomplete tensor category: If $R,S,T$ are algebra objects (sometimes also called monoid objects), $M$ is some $(S,R)$-bimodule object and $N$ is some $(R,T)$-bimodule object, then $M \otimes_R N$ carries the structure of an $(S,T)$-bimodule object (does someone have a reference to the literature for this? I don't want to prove such trivialities in my thesis).
