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.

Short Exact Sequence and Finitely Generated Modules
Let $R$ be a ring with $1$. Let
\[0\to M\xrightarrow{f} M' \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}\]
be an exact sequence of left $R$-modules.
Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M'$ is also finitely generated.
[…]

Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator
(a) Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module.
Prove that the module $M$ has a nonzero annihilator.
In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$.
Here $r$ does not depend on […]

Annihilator of a Submodule is a 2-Sided Ideal of a Ring
Let $R$ be a ring with $1$ and let $M$ be a left $R$-module.
Let $S$ be a subset of $M$. The annihilator of $S$ in $R$ is the subset of the ring $R$ defined to be
\[\Ann_R(S)=\{ r\in R\mid rx=0 \text{ for all } x\in S\}.\]
(If $rx=0, r\in R, x\in S$, then we say $r$ annihilates […]

Basic Exercise Problems in Module Theory
Let $R$ be a ring with $1$ and $M$ be a left $R$-module.
(a) Prove that $0_Rm=0_M$ for all $m \in M$.
Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$.
To simplify the […]

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 $R$.
Let $M'$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$.
Prove that $M'$ is a submodule of […]

Ascending Chain of Submodules and Union of its Submodules
Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain
\[N_1 \subset N_2 \subset \cdots\]
of submodules of $M$.
Prove that the union
\[\cup_{i=1}^{\infty} N_i\]
is a submodule of $M$.
Proof.
To simplify the notation, let us […]