Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure?

Module Theory problems and solutions

Problem 418

If $M$ is a finite abelian group, then $M$ is naturally a $\Z$-module.

Can this action be extended to make $M$ into a $\Q$-module?

 
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Proof.

In general, we cannot extend a $\Z$-module into a $\Q$-module.

We give a counterexample. Let $M=\Zmod{2}$ be the order $2$ cyclic abelian group.
Hence it is a naturally $\Z$-module. We prove that this action cannot be extended to a $\Q$-action.
Let us assume the contrary.

Then for any $x\in M$, let
\[M\ni y:=\frac{1}{2}\cdot x.\] Then we have
\[x=1\cdot x=2\cdot \frac{1}{2}\cdot x=2y=0\] in $M=\Zmod{2}$.

This is a contradiction since $M$ contains a non-zero element.
Therefore, the $\Z$-action cannot be extended to a $\Q$-action.


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