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Is there a way to characterise prime ideals in affine coordinate rings (i.e. quotients of polynomial rings). To be more specific, how can I say if principal ideals in such rings are prime or not in an elementary way?

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For $R$ a commutative unital ring, there is a bijective correspondence between the prime ideals of $R/I$ with the set of prime ideals of $R$ containing $I$ (reduced mod $I$).

For example, the prime ideals of $\mathbb C[x]/(x^2)$ are the prime ideals of $\mathbb C[x]$ which lie above the ideal $(x^2)$. The prime ideals of $\mathbb C[x]$ are of the form $(x-a)$ for any $a\in\mathbb C$, or $(0)$. The only one of these ideals which contains $(x^2)$ is $(x)$, so $\operatorname{Spec}(\mathbb C[x]/(x^2))=\{(x+(x^2))\}$ (which would usually just be written $(x)$).

In the language of schemes, this says that the closed subsets $V(I)$ of $\operatorname{Spec}(R)$ may be identified with the affine schemes $\operatorname{Spec}(R/I)$. This is in fact how we define a closed subscheme of an affine scheme.

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I guess, you've recently read the geometry of schemes by Eisenbud and Harris ;) –  Ehsan M. Kermani Jan 5 '12 at 6:36
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