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)

  • $\begingroup$ I mean, all modules in commutative algebra are (possibly not finitely generated) $\Bbb Z$-modules... $\endgroup$
    – user239203
    Commented Sep 14, 2019 at 9:49

1 Answer 1


$\mathbf Q/\mathbf Z$ is an example of a torsion $\mathbf Z$-module with zero annihilator.


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