Questions of a completely reducible module Please help to deal with the tasks of:
$1)$ Which cyclic groups are completely reducible as a $\mathbb Z$-modules?
$2)$ Which cyclic modules are completely reducible over the ring $\mathbb F[x]$, where $\mathbb F$ is the field?   
As I understand it , M - completely reducible if decomposes into a direct sum of minimal sub-modules M. However, I can't understand how to find these cyclic groups. 
I would appreciate any helpful advice!

A module is completely reducible (or semisimple) if and only if it is a direct sum of simple modules.
The simple $\mathbb{Z}$-modules are the cyclic groups $\mathbb{Z}/p\mathbb{Z}=\mathbb{Z}(p)$, where $p$ is prime, so a completely reducible $\mathbb{Z}$-module can certainly be written as
$$
\bigoplus_{p\in\mathbb{P}}\mathbb{Z}(p)^{(n_p)}
$$
where $n_p$ is some cardinal number and $\mathbb{P}$ is the set of prime numbers.
Since a cyclic $\mathbb{Z}$-module is finitely generated, we must have $n_p=0$, for all but a finite number of primes.
Since every subgroup of a cyclic group is cyclic and $\mathbb{Z}(p)\oplus\mathbb{Z}(p)$ is not cyclic, we cannot have $n_p>1$ for any $p$.
Thus the module can be written as
$$
\mathbb{Z}(p_1)\oplus\mathbb{Z}(p_2)\oplus\dots\oplus\mathbb{Z}(p_n)
\cong\mathbb{Z}(p_1p_2\dots p_n)
$$
for some distinct primes $p_1,p_2,\dots,p_n$.
Is there any fundamental difference in the case of $F[x]$?
