This is about problem $5$ in Section IV.$5$ of Hungerford's Algebra book. The question is the following:
If A' is a submodule of the right $R$-module $A$ and B' is a submodule of the left $R$-module $B$, then \frac{A}{A'} \otimes _R \frac{B}{B'} \cong \frac{A \otimes _R B}{C} where $C$ is the subgroup of $A \otimes _R B$ generated by all elements a' \otimes b and a \otimes b' with $a \in A$, a' \in A', $b \in B$, b' \in B'.
I'm just starting to somehow grasp the tensor product concept and I'm having a lot of trouble when trying to prove that some tensor product is isomorphic to a certain group.
In particular what I feel that's causing me problems is the fact that in the tensor product $A \otimes _R B$ not every element is an elementary tensor of the form $a \otimes b$ so that it is really hard for me sometimes to be able to define an appropriate map to try to produce an explicit isomorphism.
I'm not sure if the universal property could be of help here, since I can try to define a map \varphi : \frac{A}{A'} \times \frac{B}{B'} \longrightarrow \frac{A \otimes _R B}{C} by \varphi(a + A', b + B') := a \otimes b + C
But I've been trying without getting anywhere. Any hints or advice on how to approach this exercise would be very much appreciated. Thank you.