Some questions on Proof of Structure Theorem I understand the general idea of the proof of Structure Theorem for finitely generated modules over a principal ideal domain but I found it quite difficult to follow some lines of reasoning in the proof. I have spent hours tried to think about it and look for other sources but still unable to fully convince myself. Any detailed helps are truly appreciated.
The Theorem is stated as follows:  

Let $M$ be a finitely generated module over a principal ideal domain $R$. Then there exist elements $d_1, d_2, ..., d_k \in R$ satisfying $d_1\mid d_2\mid\cdots\mid d_k$ such that $$M\cong R/(d_1)\oplus R/(d_2)\oplus \cdots\oplus R/(d_k).$$

Proof:
Since $M$ is finitely generated (by $k$ elements say), then $M\cong R^k/N$. But $N\le R^k\Rightarrow N\cong R^s$ for some $s\le k$. Let $\zeta:R^s\to N$ be an isomorphism. This gives a homomorphism $\varphi:R^s\to R^k$ with $Im(\varphi)= N, ker(\varphi)=\{0\}$. Let $A\in M_{k\times s}(R)$ be the matrix of $\varphi$ with respect to the standard bases for $R^s$ and $R^k$. Then A is equivalent to a matrix 
$$D=\begin{bmatrix}d_1 & 0 & ... & & 0\\0 & d_2 & 0 & ... & 0\\ & &.& & \\ & &:& & \\0 & 0 & 0 & ... & d_s\\0 & 0 & 0 & ... \\0 & 0 & 0 & ... \end{bmatrix},$$ where $D = XAY$, for $X\in M_{k\times k}(R)$ invertible and $Y\in M_{s\times s}(R)$ invertible. So $D=[\varphi]_{C,B}$ for some bases $B$ for $R^s$, $C$ for $R^k$. Let $C=\{f_1,f_2,...,f_k\}$ be a basis for $R^k$, then $\varphi(B)=\{d_1f_1,d_2f_2,...,d_kf_k\}$ is a basis for $N\subseteq R^k$. So we have $$R^k=<f_1>\oplus<f_2>\oplus...\oplus<f_k>,$$ $$N=<d_1f_1>\oplus<d_2f_2>\oplus...\oplus<d_sf_s>\oplus<0f_{s+1}>\oplus...\oplus<0f_k>$$ where $d_i:=0$ for $s<i\le k$. So, 
$$R^k/N\cong\frac{<f_1>}{<d_1f_1>}\oplus\frac{<f_2>}{<d_2f_2>}\oplus...\oplus\frac{<f_k>}{<d_kf_k>}.$$But for any $v\in R^k\setminus \{0\}$, by First Isomorphism Theorem, $<v>/<dv>\cong R/(d)$. So $R^k/N\cong R/(d_1)\oplus R/(d_2)\oplus ...\oplus R/(d_k)$, $d_1|d_2|...|d_{s}|d_{s+1}=0|...$. (QED)
My questions are:
1. Why is $ker(\varphi)=\{0\}$? I could not find any convincing explanation.
2. I have been told that $C$ is given by columns of $Y$ and $B$ is given by columns of $X^{-1}$. I have some understanding of change of bases but how can we show it in a more rigorous way?
3. The part that I struggle the most are $R^k=<f_1>\oplus<f_2>\oplus...\oplus<f_k>$ and $N=<d_1f_1>\oplus<d_2f_2>\oplus...\oplus<d_sf_s>\oplus<0f_{s+1}>\oplus...\oplus<0f_k>$, what does <> represent, why are we taking their direct sum and how do we derive them?
4. Lastly, why can we just divide $R^k/N\cong\frac{<f_1>}{<d_1f_1>}\oplus\frac{<f_2>}{<d_2f_2>}\oplus...\oplus\frac{<f_k>}{<d_kf_k>}$ on both sides, how can I explain/convince myself rigorously.
5. I have consulted other sources, in the final step, it is given that the map $\psi:R^k\to R/(d_1)\oplus R/(d_2)\oplus ...\oplus R/(d_k)$ given by $\psi(\sum_{i=1}^{k}r_if_i)=(r_1+(d_1),...,r_k+(d_k))$ is a homomorphism of R-modules. The result follows from FIT since $\psi$ is surjective and $ker(\psi)=N$. How can we show $\psi$ is surjective and $ker(\psi)=N$?
6. Why if $d_i=0$, then $d_i=0$ for all $j\ge i$? And why if $d_i$ is a unit, then $d_j$ is a unit for all $j\le i$?
Many thanks in advance!
 A: Some ideas:
The homomorphism $\,\phi\,$ described there could be thought of as an embedding one: note that $\,N\cong R^s\le R^k\,$ , so one can embed the former in the latter. 
Of course, this embedding is usually far from being unique. From here  $\,\ker\phi=0\,$ is immediate.
The notation $\,\langle x\rangle\,$ usually denotes the cyclic (module, group, etc. ) generated by $\,x\,$ . In this case for example, $\,\langle f_i\rangle\,$ denotes the cyclic $\;R$-module  generated by $\,f_i\,$ in $\,R^k\,$ , i.e.:
$$\langle f_i\rangle:=\{rf_i\;;\;r\in R\}$$
Observe that any finitely generated $\,R$-module $\;M\;$ is the set of all the $\,R$-linear combinations of a finite number of elements of $\,M\,$ , and from here the term "basis" .
The dividing thing is just taking the quotient module...Apparently you already proved before this that the quotient takes that form (this is (5) in your questions):
$$\bigoplus_{i=1}^k\langle f_i\rangle/\bigoplus_{i=1}^k\langle d_if_i\rangle \cong\bigoplus_{i=1}^k \langle f_i\rangle/\langle d_if_i\rangle$$
Your questions about $\,\psi\,$ could probably be partially answered if you read carefully about the CRT = Chinese Remainder Theorem , which has a very similar homomorphism there, yet directly: clearly $\,\psi\,$ is surjective since any element in the direct sum of quotient modules on the right is precisely of the form $\;r_j+(d_j)\;$ , for some $\,r_j\in R\,$ , and you can take an element in $\,R^k\,$ with these $\,r_j$-s as corresponding coefficients of the basis $\,\{f_1,...f_k\}\,$ . That $\,N=\ker\psi\,$ is even more trivial (Sorry, I'm not teasing you, honest) if one understand what's the role of each thing (and this is why I thought subdividing the question in several, focused parts could help...)
