Finitely generated modules over PIDs and vector spaces

Question

I am trying to understand and complete the proof of the following statement: Let $A$ be a module over the ring $\mathbb{C}[x]$. Since $\mathbb{C}\subset\mathbb{C}[x]$, $A$ is a $\mathbb{C}$-vector space. List all the $\mathbb{C}[x]$-modules $A$ up to isomorphism with $dim_{\mathbb{C}}A=1,2,3$. What about $dim_{\mathbb{C}}A=4$?

Incomplete proof: $A$ is a finitely generated over $\mathbb{C}[x]$, which is a PID. So by the structure theorem: $A\cong\mathbb{C}[x]\oplus\cdots\oplus\mathbb{C}[x]\oplus\mathbb{C}[x]/(p_1)\oplus\cdots\oplus\mathbb{C}[x]/(p_m)$ with $p_i$ a power of an irreducible.

As I understand it, we need to now use the dimension of $A$ as a $\mathbb{C}$-vector space to conclude that:

1. For $dim_{\mathbb{C}}A=1$ the module isomorphism is an isomorphism of $\mathbb{C}$-vector spaces and so for the right side consists of one factor, which is a field. Since it is a field $p_1$ must be irreducible, i.e. $A\cong \mathbb{C}[x]/(p_1)\cong \mathbb{C}$ since only linear polynomials are irreducible over $\mathbb{C}$. Is this correct? And if so why is the module isomorphism a vector space isomorphism?
2. For $dim_{\mathbb{C}}A=2$ we have $A\cong\mathbb{C}[x]/(p_1)\oplus\mathbb{C}[x]/(p_2)$ or $A\cong\mathbb{C}[x]/(p_1^2)$ and for $dim_{\mathbb{C}}A=3$ we have $A\cong\mathbb{C}[x]/(p_1)\oplus\mathbb{C}[x]/(p_2)\oplus\mathbb{C}[x]/(p_3)$, $\mathbb{C}[x]/(p_1)\oplus\mathbb{C}[x]/(p_2^2)$ or $\mathbb{C}[x]/(p_1^3)$. But what how can we conclude this? Since $p_{1,2}^2$ and $p_1^3$ are not irreducible and so $\mathbb{C}[x]/(p^{2,3})$ are not fields. Why can we then use the dimension (a property of vector spaces) to argue this?
3. I suppose that for $dim_{\mathbb{C}}A=4$ we have $A\cong\mathbb{C}[x]/(p_1)\oplus\mathbb{C}[x]/(p_2)\oplus\mathbb{C}[x]/(p_3)\oplus\mathbb{C}[x]/(p_4)$, $A\cong\mathbb{C}[x]/(p_1)\oplus\mathbb{C}[x]/(p_2)\oplus\mathbb{C}[x]/(p_3^2)$, $A\cong\mathbb{C}[x]/(p_1^2)\oplus\mathbb{C}[x]/(p_2^2)$, $A\cong\mathbb{C}[x]/(p_1)\oplus\mathbb{C}[x]/(p_2^3)$ or $A\cong\mathbb{C}[x]/(p_1^4)$. But I have the same confusion as above.

I would be grateful for any help you can offer.