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Question: Let $E$ be a torsion abelian group. Prove that $E$ has exactly one $\hat{\mathbb{Z}}$-module structure, and that the scalar multiplication $\hat{\mathbb{Z}} \times E \rightarrow E$ defining this module structure is continuous, if $E$ is given the discrete topology.

Attempt: So I have already shown that

$\hat{\mathbb{Z}} \times E \rightarrow E : ((a_{n})_{n > 0}, e) \mapsto a_{m}e$

defines a module structure where $\text{ord}(e) = m$ by verifying the axioms. Also, I have already shown that the module structure is continuous if $E$ is given the discrete topology.

However I am struggling with proving the uniqueness. If anybody has a hint for this I would appreciate that. This is the first question I have ever encountered about module structures, so I am unaware of any theorems that could be applicable. This question has been asked before but there have been no responses about the uniqueness part of the question.

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    $\begingroup$ $E$ has a unique $\mathbb Z$-module structure and $\mathbb Z$ is dense in $\hat{\mathbb Z}$. $\endgroup$ Commented Jun 11 at 11:07
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    $\begingroup$ You can reformulate a $\hat{\mathbb{Z}}$-module structure on $E$ as a ring morphism $\hat{\mathbb{Z}}\to \operatorname{End}(E)$. $\endgroup$ Commented Jun 11 at 11:52
  • $\begingroup$ Thank you for your comment @CaptainLama. I looked into that reformulation and I understand that better. But I am still confused as to how this would show the uniqueness. Could you explain it please? $\endgroup$ Commented Jun 14 at 10:30

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