Construct a unique simple submodule

  • #1
TMO
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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?
 

Answers and Replies

  • #2
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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?
 

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