This is a standard problem I've seen, but I am new to tensors. If I could see the proof I think it would help me greatly to compute tensors in general.

The problem statement here is from Hungerford GTM section IV.5.2

Show $\displaystyle Z_m \otimes Z_n \cong Z_c$ where c is gcd(m,n).

(everything is regarded as an add. abelian group-- the tensor is over Z)

A lead question is to show $\displaystyle A \otimes Z_m \cong A/mA$, which I think I managed with the mapfsending $\displaystyle a \otimes (n+mZ) \mapsto (an+mA)$ showing the kernel was 0 and was surjective. For what it's worth, as a bijection this implies elements of $\displaystyle A \otimes Z_m$ are required to have the form simply $\displaystyle a \otimes 1$-- whether or not this will help with the problem stated above has not been something I can reconcile on my own.

Any insights would be appreciated greatly.