Let $M$ be a finitely generated torsion module over a PID $D$. Suppose $\varphi:M\to{N}$ is a module epimorphism and suppose that the invariant factor ideals of $M$ are $(d_1)\supseteq{(d_2)}\supseteq...\supseteq(d_s)$.
Show that $N$ is a finitely generated torsion module with invariant factor ideals $(c_1)\supseteq{(c_2)}\supseteq...\supseteq(c_t)$ such that $t\le{s}$ and $c_t|d_s,c_{t-1}|d_{s-1},...,c_1|d_{s-t+1}$
Here is my attempt:
By Structure theorem, there exist $z_1,z_2,...,z_s$ such that $$M=Dz_1\oplus...\oplus{Dz_s}$$ and $Ann{z_i}=(d_i)$;
because $\varphi$ is surjective, $N=\sum_{i=1}^s D\varphi(z_i)$ and so $N$ is finitely generated.
Given $y\in{N}$ there exists $x\in{M}$ such that $\varphi(x)=y$; thus $d_sy=\varphi(d_sx)=\varphi(0)=0$ and since $d_s\neq{0}$ it follows that $N$ is indeed a torsion module.
For $p$ a prime, let $M_p$ be the p-component of M, the submodule of $M$ consisting of the elements annihilated by some power of $p$. I know that $$M=torM=M_{p_1}\oplus...\oplus{M_{p_h}}$$ for a finite number of distinct primes and it is easy to show that $\varphi(M_p)\subseteq{N_p}$. Then, since by surjectivity $$N=\varphi(M_{p_1})+...+\varphi(M_{p_h})$$ and the $N_{p_i}$'s are independent submodules of $N$, it follows that $$N=\varphi(M_{p_1})\oplus...\oplus\varphi(M_{p_h})$$.
Now I think it is sufficient to prove the theorem for a primary module $M=M_p$ because the invariance theorem will guarantee the divisibility requirement.
Finally, let $M$ be a primary module so that $$M=Dz_1\oplus...\oplus{Dz_s}$$ with $annz_i=(p^{e_i})$ and $e_1\le{e_2}\le...\le{e_s}$.
We have $$N=\varphi(M_p)\subseteq{N_p}\subseteq{N}$$ therefore $$N=N_p$$ and so $$N=Dy_1\oplus...\oplus{Dy_t}$$ with $anny_i=(p^{f_i})$ and $f_1\le{f_2}\le...\le{f_t}$. Since $p^{e_s}y=0$ $\forall{y}\in{N}$ it follows that $p^{e_s}\in{ann{y_t}}=(p^{f_t})$ and so $f_t\le{e_s}$.
To conclude the proof it is enough to show that $f_{t-1}\le{e_{s-1}}$ since we can repeat the argument for the other indices, but unfortunately I am not able to do so.
Is this the right approach? If so, can anyone help me ending the proof?
Thank you.