This is follow up question Understanding the Details of the Construction of the Tensor Product
If $M$ and $N$ are 2 A-modules, we define $Z = A^{M\times N}$ as module generated by $M\times N$. Then $D$ is sub-module of $Z$ defined with some relations as described above.
I am not understanding why these relations are like that. It can be simplified as -- $(v+v_1,w) - (v,w) - (v_1,w) \implies (0,-w)$ and similarly for other 3 conditions
As per my understanding, $Z/D$ is set of cosets of $D$. If $(v,w)$ belongs to $Z$ then its corresponding coset will be $(v,w) + D$. What is meant by image of $(v,w)$ ?