Characterize the irreducible torsion modules over a PID

Let R be a principal ideal domain and let M be a torsion R-module. Prove that M is irreducible if and only if M = Rm for some nonzero element m \in M where the annihilator of m in R is a nonzero prime ideal P.

Suppose M is an irreducible, torsion R-module. By this previous exercise, we have M \cong_R R/I where I \subseteq R is a maximal ideal. Since R is a principal ideal domain, in fact I = (p) for some prime element p \in R. Let \psi : R/(p) \rightarrow M be an isomorphism, and let m = \psi(1 + (p)). Now M = Rm, and moreover \mathsf{Ann}_R(m) = \mathsf{Ann}_R(1 + (p)) = (p).

Conversely, consider the module M = Rm, where \mathsf{Ann}_R(m) = (p) is a prime ideal. Define \psi : R \rightarrow (m)_R by r \mapsto r \cdot m. Certainly we have \mathsf{ker}\ \psi = \mathsf{Ann}_R(m) = (p). By the First Isomorphism Theorem for modules, we have \overline{\psi} : R/(p) \rightarrow M an isomorphism. Again by this previous exercise, R/(p) is irreducible.

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