Isomorphism between quotient of ring and principal ideals Let $R$ be a principal ideal domain and suppose that $u,u'\in R$ are such that $R/(u)\cong R/(u')$ as $R$-modules, where $(u)$ denotes the ideal generated by $u$. Is it true that $u=\alpha u'$, where $\alpha\in R$ is invertible?
I have tried to prove this unsuccesfully, but it ought to be true, in order to have that the order of cyclic module over a PID is well-defined up to association with a unit.

If $R/(u)\simeq R/(u')$ as $R$-modules, then they have the same annihilator (why?), so $(u)=(u')$, and thus $u=\alpha u'$ with $\alpha\in R$ invertible.
Edit. In this answer $R$ is an integral domain.
