# Can $\Z$-Module Structure of Abelian Group Extend to $\Q$-Module Structure? ## 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? Add to solve later

## 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. Add to solve later

### More from my site

#### You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

###### More in Module Theory ##### Submodule Consists of Elements Annihilated by Some Power of an Ideal

Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$....

Close