Let $X$ be any topological space, and let $R$ be any commutative ring with identity (in particular, a $\mathbb{Z}$-module). Let $C_c^{\infty}(X,R)$ be the $R$-module of locally constant functions with compact support from $X$ to $R$. I was wondering if $$C_c^{\infty}(X,\mathbb{Z})\otimes_{\mathbb{Z}}R\cong C_c^{\infty}(X,R),$$ as $R$-modules, by means of the canonical $R$-linear map $f\otimes r\mapsto rf$. The surjectivity (if I'm correct) is easy, but I could not prove the injectivity. Probably, I'm missing something elementary, sorry if that is the case.
(While trying to understand a paper by Mazur, I needed to prove this when $X$ is a profinite space and when $R$ is a field of characteristic zero, which I proved but using the fact that $X$ is profinite, by reducing to the case when $X$ is finite and discrete, which is easy. I read in another paper that the above is true so I tried to prove it, unsuccesfully.)
When I'm trying to prove that something is isomorphic to a tensor product, one thing that works a lot of the time is constructing an inverse map. So you can define a map $\phi:C_c^{\infty}(X,R) \rightarrow C_c^{\infty}(X,\mathbb Z) \otimes_{\mathbb Z}R$ like this:
Given $f: X \rightarrow R$, taking finitely many values $c_1, ... , c_n$, let $f_i \in C_c^{\infty}(X,R)$ be the characteristic function of $f^{-1}(c_i)$. Then $f = c_1 f_1 + \cdots + c_nf_n$, and you can define
$$\phi(f) = f_1 \otimes c_1 + \cdots + c_n \otimes f_n$$ where now you are thinking of $f_i$ as an element of $C_c^{\infty}(X,\mathbb Z)$.
You have to do a little work to show that $\phi$ is an abelian group homomorphism. Once you do, showing that $\phi$ is inverse to the map you defined is easy.