I am doing some work on $Tor$, and maybe Odin too (joke!). Been calculating it for some and it's interesting to say the least. Well I have reached a step I feel intuitively is the case but I am not certain so I bring it up here to see if my hunch is correct.
I have $$\mathbb{Z}_m\otimes \mathbb{Z}_n/p(\mathbb{Z}_m\otimes\mathbb{Z}_n)$$ as a quotient module over the ring $\mathbb{Z}_m$ here to calculate it, I can probably prove my idea through brute force (which I haven't as I want to see if my hunch is correct first), but if it is correct I'd love a shorter form of proof. My idea would be that this is isomorphic to $\mathbb{Z}_p$ in general, though I am fairly certain also in my gut that there probably are some restrictions that makes this not universal. The first natural one is that $p<m$ by the nature of the ring and module. A second one is perhaps it has to be a prime but not certain.
Is my intuition correct? Is also my gut correct that there must be certain limitations to this? What would be the quickest way to prove either one if correct? If not why does it fail? I know, quite a bit of questions, just came to me so I am still working on it. Thanks in advance.
