Construct a unique simple submodule

[TMO]

Problem. Let $p$ be a prime integer. Let $Z_p^\infty$ be the set of complex numbers having order $p^n$ for some $n \in \mathbb{N}$, regarded as an abelian group under multiplication. Show that $Z_p^\infty$ has an unique simple submodule.

Attempted solution. The collection of all elements of order $p$ of $Z_p^\infty$ has the structure of an abelian cyclic group. Now how do I also show that it is closed under scalar multiplication? Would showing that it is closed under scalar multiplication suffice to prove that it is a submodule?

 

[fresh_42]

Problem. Let $p$ be a prime integer. Let $Z_p^\infty$ be the set of complex numbers having order $p^n$ for some $n \in \mathbb{N}$, regarded as an abelian group under multiplication. Show that $Z_p^\infty$ has an unique simple submodule.

Attempted solution. The collection of all elements of order $p$ of $Z_p^\infty$ has the structure of an abelian cyclic group. Now how do I also show that it is closed under scalar multiplication? Would showing that it is closed under scalar multiplication suffice to prove that it is a submodule?

What is the module structure of $\mathbb{Z}_p^\infty$, means what is the ring of scalars?