# Homework Help: [Module theory] Prove that something forms a left R module.

1. Apr 23, 2012

1. The problem statement, all variables and given/known data
Suppose that R and S are two rings, M, is a (R-S) bi-module and N is a left R-module. Show that $M \otimes N$ has the structure of a left S-module.

3. The attempt at a solution

Well, $M\otimes N$ is an Abelian group, so it's enough that I define a scalar product on $M\otimes N$. I'm thinking of defining:
$s.\sum_{i=1}^t{x_i\otimes y_i} = \sum_{i=1}^t{sx_i\otimes y_i}$
Now I'm a bit clueless about how I should show that this scalar multiplication is well-defined. I know that I should suppose $\sum_{i=1}^t{x_i\otimes y_i} = \sum_{i=1}^{t'}{x'_i\otimes y'_i}$ and then show that $\sum_{i=1}^t{sx_i\otimes y_i} = \sum_{i=1}^{t'}{sx'_i\otimes y'_i}$ but I don't know how I should do that. I'm looking for a nice map from $F$ to $F/K = M\otimes N$ that does the trick but nothing good comes to my mind now. Any ideas would be appreciated greatly.