Fix a prime number $p$. I think the module I am working has a standard notation but I am using some notes that are not following any textbooks I know of. Let $J = \{ \frac{l}{p^m} + \mathbb{Z} : l \in \mathbb{Z} , m \in \mathbb{N} \}$ Let $M = H \times K$ where $H,K$ are canonically identified with submodules of $J$.
The problem I am working on has a couple parts and the first part is easy it just requires one to show the mapping $ \phi :M \rightarrow M$ given by $(x,y) \rightarrow (x,y+px)$ is a bijective homomorphism. The next part is where I start to get confused:
Set $H^* = \phi(H) $ how do we show that $H$ and $H^*$ are both direct factors of $M$ but $H \cap H^*$ is not a direct factor of $M$?
After trying a couple ideas I keep getting that $H \cap H^* = \{ (x,0) : x \in J \} $. I thought the way we show that $H$ and $H^*$ are direct factors are to show that $ M = H \oplus H^*$ but I am confused because we would need $H \cap H^* = 0$. Any advice would be greatly appreciated