Let be a principal ideal domain and let be a torsion -module. Prove that is irreducible if and only if for some nonzero element where the annihilator of in is a nonzero prime ideal .
Suppose is an irreducible, torsion -module. By this previous exercise, we have where is a maximal ideal. Since is a principal ideal domain, in fact for some prime element . Let be an isomorphism, and let . Now , and moreover .
Conversely, consider the module , where is a prime ideal. Define by . Certainly we have . By the First Isomorphism Theorem for modules, we have an isomorphism. Again by this previous exercise, is irreducible.