I am currently attempting the following:
Find (cyclic) $\mathbb{Z}$-modules $M, N, P$ and an injective homomorphism $f: M \rightarrow N$ s.t. $g: M \otimes_{\mathbb{Z}} P \rightarrow N \otimes_{\mathbb{Z}} P$ is not injective.
I feel like there should be simple examples to show this, but I just can't seem to find anything. I'm guessing that if I was to replace $\mathbb{Z}$ with a field, then the injectivity of $f$ would imply the injectivity of $g$, but I still can't seem to be able to figure out an example for this case!
Any hints / nudges would be greatly appreciated!