$\newcommand{\C}{\mathbb{C}}$I'm a bit confused by quotient rings, and I want to know the general technique for describing their ideals.
For example, say I have the ring $\C[x]/(x^3)$
If we consider the homomorphism
$\C[x] \to C[x]/(x^3)$
then the kernel of this map is $(x^3)$, so if we consider all ideals $I$ containing $x^3$, we see that those are $x^1, x^2, x^3$ so can I conclude that my ring has 3 ideals? What are those ideals though?