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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

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Might as well collect the comments to a partial answer. The remaining details will depend on the details of the choice of $H$ and $K$.

The meaning of "$H$ is a direct factor of $M$" simply means that $M=H\times N$ for some other submodule $N$. Here we are given that $M=H\times K$. As $\phi$ is an automorphism of $M$, we see that also $M=\phi(H)\times\phi(K)$. This is hopefully not surprising: any element $(x,y)$ can be written as $$ (x,y)=(x,px)+(0,y-px), $$ where the first term is in $\phi(H)$ and the latter in $\phi(K)=K$. Furthermore, it is easy to show that $\phi(H)\cap\phi(K)=0$.

A probable cause of difficulty for you is that you seem to have a wrong idea about the intersection $H\cap \phi(H)$. Note that $H=\{(x,0)\mid x\in H\}$ when viewed as a subset of the (external) direct product. Therefore $H^*=\phi(H)=\{(x,px)\mid x\in H\}$. So for an element of $H$ to belong to the intersection $H\cap H^*$ it is necessary and sufficient that $px=0$. Therefore $$ H\cap H^*=\{x\in H\mid px=0\} $$ as a subset of $H$. The task at hand is to prove that $M$ cannot be written in the form $M=(H\cap H^*)\times N$ for any submodule $N$. I cannot tell for sure, but my guess is that you can locate an element $m\in M$ such that $pm\in H\cap H^*$, but $m\notin H\cap H^*$. It will be very difficult to write $m$ as a sum of something from $H\cap H^*$ and something from another submodule that intersects trivially with $H\cap H^*$. Hopefully this gets you started.

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