Let $M,N$ be two torsion free modules over Dedekind domain $O$ with same rank. Suppose $M\subset N$. Then one can talk about module index $[M:N]_O$. Module index is defined in terms of local field automorphism $l_p:M_p\to N_p$ which exists by PID. Then $[M_p:N_p]_{O_p}=det(l_p)O_p=p^{k_p}O_p$. Then $[M:N]_O=\prod_{p\in Spec(O)}p^{k_p}$.
$\textbf{Q:}$ If $[M:N]_O=O$ for $N\subset M$, do I know $M=N$? It is clear that $M\cong N$. Do I know this morphism is $M=N$ morphism?