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$\newcommand{\C}{\mathbb{C}}$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?

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    $\newcommand{\C}{\mathbb{C}}$The ideals are $$ \C[x]/(x^{3}),\ \text{which is the whole ring,} \quad (x)/(x^{3}), \quad (x^{2})/(x^{3}), \quad (x^{3})/(x^{3}),\ \text{which is the trivial ideal}. $$ – 2017-02-18
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    Ohh, thank you. I've never seen this notation before. I thought they should all be in the form (x)? – 2017-02-18
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    Which notation are you referring to? – 2017-02-18
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    You can also write $(x)/(x^{3}) = (x + (x^{3}))$, if you prefer. – 2017-02-18
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    @AndreasCaranti: How does one show that those are *all* the ideals? In general, if $p : A \to A/I$ (with $A$ commutative) is the natural projection, is it possible to list the ideals of $A/I$? For sure, if $J$ is an ideal in $A$, then $p(J)$ is an ideal in $A/I$, but these seem not to be the only ones. Plus, if $K$ is another ideal in $A$, it may happen that $p(I) = p(J)$, so how to list these ideals just once, without repetitions? – 2017-02-18
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    @AlexM. It follows from the [Lattice Isomorphism Theorem](https://crazyproject.wordpress.com/2010/04/05/the-lattice-isomorphism-theorem/). The ideals of $A/I$ are of the form $J/I$ where $J$ is an ideal of $A$ with $I \subseteq J \subseteq A$. – 2017-02-18
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    @SpamIAm, thx, I second that. – 2017-02-19
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    @SpamIAm: Thank you, your comment (with a bit of extra meat on it) should be turned into a full answer, since it also addresses the OP's question (which, incidentally, was not about $\Bbb C[x] / (x^3)$, that was just an example, but about quotients of commutative rings in general). Also, a better link might be https://en.wikipedia.org/wiki/Isomorphism_theorem#Third_isomorphism_theorem_2. – 2017-02-19
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    Aren't those: $(x)/(x^3),(x^2)/(x^3)$ just $(x)$ and $(x^2)$? – 2017-02-21

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