Let \(R\) be a principal ideal domain, \(p\in R\) a prime, and \(M\) an \(R\)-module. \(M\) is \(\gens{p}\)-primary if every \(m\in M\) is annihilated by some power of \(p\): for each \(m\in M\) there is \(n\geq 1\) with \(p^n m = 0\).
For any \(R\)-module \(M\), the \(p\)-primary component \(M[p^\infty] \coloneqq \theset{ m \in M \st p^n m = 0 \text{ for some } n\geq 1}\) is a submodule, and \(M\) is \(\gens{p}\)-primary if and only if \(M[p^\infty]=M\).
For $R$ a PID and $p \in R$ prime, when is an $R\dash$module $\gens{p}\dash$primary?