Isomorphism between quotient of ring and principal ideals

Let R be a principal ideal domain and suppose that$u,{u}^{\prime}\in R$ are such that $\frac{R}{\left(u\right)}\stackrel{\sim}{=}\frac{R}{\left({u}^{\prime}\right)}$ as R-modules, where (u) denotes the ideal generated by u. Is it true that $u=\alpha {u}^{\prime}$ , where $\alpha \in R$ is invertible?

Let R be a principal ideal domain and suppose that