Tensor product as module - standard terminology? In Dummit and Foote,2004; pg 375 they ask you to show that $\Bbb{C}\otimes_{\Bbb{C}}\Bbb{C}$ is a left $\Bbb{R}$-modules without clarification of the action. My question is; when something like this is posed is it standard to assume the action is:
$$r\left(\sum_{finite}a\otimes b\right)=\sum_{finite}(ra)\otimes b$$
in this case with $r\in \Bbb{R}$ and $a\otimes b\in \Bbb{C}\otimes_{\Bbb{C}}\Bbb{C}$. If this is not standard/what they mean then what is? 
I am interested in the general case of e.g. $M\otimes_R N$ as a $S$-modules - not just the case presented here as an example. Here $R$ and $S$ are rings with $1$ and are possibly non-commutative. $M$ and $N$ have to be at least $R$-modules.

Yeah, that's typically the natural action to consider. But then you have to prove that it is a valid left action. 
In general, to prove something is a left $R$-module, it just has to have some action of $R$ defined on the left. You get to pick that action, and usually it turns out to be a very natural choice. In the general case, part of the exercise is deciding what the "natural action" is. 
