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## Main Question or Discussion Point

Let's suppose that I have an element ##e## of order ##p## in the group of complex numbers whose elements all have order ##p^n## for some ##n\in\mathbb{N}## (henceforth called ##K##), and the module generated by ##(e)## is irreducible.

How do I show that the injective hull of the module generated by ##(e)## is in fact, equal to ##K##?

How do I show that the injective hull of the module generated by ##(e)## is in fact, equal to ##K##?

**Attempted Work.**I was told that this submodule generated by ##(e)## is isomorphic to ##\mathbb{Z}[p^{-1}]##. I don't know how to proceed from there... but I think it involves showing that two maps are surjective.