tensor product of one-one linear maps 
Let $f_1:M_1\to M_1'$ and $f_2:M_2\to M_2'$ be one-one linear maps, where $M_1,M_1',M_2,M_2'$ are free $R$-modules. Is $f_1\otimes f_2: M_1\otimes M_2\to M_1'\otimes M_2'$ defined by $$(f_1\otimes f_2)(x\otimes y)= f_1(x)\otimes f_2(y)$$
a one-one linear map?

If $R$ is a field then this is true. This can be prove using the fact that any linearly independent subset of a vector space can be extended to a basis of the given vector space. Does a similar result - Any linearly independent subset of a module can be extended to a basis of the module - also holds for an arbitrary module?
 A: Unfortunately a result like Any linearly independent subset of a module can be extended to a basis of the module does not generally hold, not even for free modules, see this question and answer.
The result more generally holds if $M'_1$ and $M_2$ or $M_1$ and $M'_2$ are flat, i.e. tensoring with these modules is an exact functor. In more down to earth terms: the tensor product is right exact, this means that given an epimorphism $P \to P'$ of $R$-modules, then tensoring with an arbitrary $R$-module $Q$ yields an epimorphism $P \otimes Q \to P' \otimes Q$. Tensoring is not in general left-exact, i.e. monomorphism are not necessarily taken to monomorphisms.
For this problem, notice that $f_1\otimes f_2$ can be decomposed as
$$
M_1 \otimes M_2 \overset{f_1 \otimes \text{id}}{\longrightarrow} M_1' \otimes M_2 \overset{\text{id} \otimes f_2}{\longrightarrow} M_1' \otimes M_2'.
$$
if we assume $M_1', M_2$ to be flat, then, by the above discussion, $f_1 \otimes \text{id}_{M_2}$ and $\text{id}_{M_1'} \otimes f_2$ are monomorphisms, hence so is $f_1\otimes f_2$.
Decomposing $f_1 \otimes f_2$ as $(\text{id}_{M_2'} \otimes f_1) \circ (f_2 \otimes \text{id}_{M_1})$ yields the same result for $M_2', M_1$ flat.
Free modules are flat, as are projective modules, so this result is true for huge (but admittedly very 'tame') classes of $R$-modules. I will try to think of some counterexamples for non-flat modules and edit this later.
Hope this helps!
