It is well known fact that if $ M $ is a non-zero finitely generated module over $ R $, where $ R $ is PID, then it can be represented as $$ M=\oplus_{i=1}^n{M_{p_i}}, $$ where $ p_i\in R $ are primes.
But what happens if $ M $ is not finitely generated? As far as I know, this assumption is needed to show that $ \mathrm{Ann}_R(M)\ne 0. $ So, is it possible that annihilator is trivial in this case? And are there any analogues to the statement above? (for example, that $ M $ is infinite direct sum of $ p_i $-modules)
$\mathbf Q/\mathbf Z$ is an example of a torsion $\mathbf Z$-module with zero annihilator.