This is a refined version of a question I asked days ago and have no answers yet. I am completely illietarte in algebra so, please, don't kill but explain.

The question is all in the title: is there a complete decomposition of an arbitrary PID module into a direct sum of indecomposable submodules? I understand that in general uniqueness and the cardinality of the set of indecomposable submodules are disputable. I just need the existence. Thank you.

**Edit** Thanks for all comments. Maybe I am close to understanding where my problem is. I think what I can always do is the following (I put it for the case of $\mathbb{Z}^\infty$ for simplicity). For every $x\in\mathbb{Z}^\infty$ such that $\mbox{gcd}(\{x_n\})=1$ the cyclic submodule $\mathbb{Z}x$ is maximal. Two such submodules either coincide or intersect trivially. Moreover, there exists by Zorn a minimal set $\{x_i\}_{i\in I}$ such that
$$
\mathbb{Z}^\infty=\sum_{i\in I}\mathbb{Z}x_i.
$$
However, this sum is still not direct, because there may be a proper submodule $\mathbb{Z}y$ with $y\in\mathbb{Z}x_j$ such that
$$
\mathbb{Z}y\in\sum_{i\in I\setminus\{j\}}\mathbb{Z}x_i.
$$
For a general PID module you may not have maximal cyclic submodules but infinitely growing sums of cyclic submodules, but the principle is apparently the same. This is too bad :(