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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$\begingroup$ There are certainly tests for whether ideals are prime. You can find some on page 209 of Ideals, Varieties and Algorithms by Cox, Little and O'Shea. It's certainly implemented in various computer algebra systems (such as Macaulay2). $\endgroup$– Karl SchwedeCommented Jan 4, 2012 at 15:04
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$\begingroup$ Or rather, in that book you can find some discussion of history and references. $\endgroup$– Karl SchwedeCommented Jan 4, 2012 at 15:24
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$\begingroup$ If you know your ring is a UFD, then your question reduces to checking if the generator of your principal ideal is irreducible. Also note that in a UFD every prime ideal of height 1 is principal. $\endgroup$– Mahdi Majidi-ZolbaninCommented Jan 4, 2012 at 16:42
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Let $R$ be a commutative ring and let $S=R[X_1,...,X_n]$.
If a principal prime ideal in the coordinate ring $A$ is represented by $f \in S$, a necessary condition is that $f$ is irreducible.
In order to generalize, let $\kappa: S \to A$ be an epimorphism. An ideal $\mathfrak{p}$ from $A$ is prime iff $P := \kappa^{-1}(\mathfrak{p})$ is prime in $S$. This reduces the problem to determing the prime ideals in $S$. Such a characterization is given in a paper (link) of Ferrero dated from 1997:
First he generalizes the notion of irreducibility (cf. Def. 2.4, 1.7), called "complete irreducibility". For an prime ideal $Q$ in $R$ and a sequence of polynomials $f_i \in R[X_1,...,X_i]$, $1 \le i \le n$, he defines an ideal $[Q,f_1,...,f_n]$ (that is simply either $(f_1,...,f_n)$ or $S$ if $R$ is a field). Then
An ideal $P$ in $S$ with $P \cap R = Q$ is prime iff there are completely irreducible polynomials $f_i \in R[X_1,...,X_i]$ such that $P=[Q,f_1,...,f_n]$.
Remarks: 1) A description of the primes in $S$ that takes the height into account can be found in Eisenbud: Commutative Algebra, Exercise 13.6.
2) If the coordinate ring is known to be Cohen-Macaulay, Serre's criterion may be of interest (cf. Eisenbud, Theorem 18.15).
3) The OP is primarily interested in principal ideals in $A$. By writing $A=S/I$ with $I= (g_1,...,g_{k-1})$ and $\mathfrak{p} = (\bar{g}_k)$, one finds $P = \kappa^{-1}(\mathfrak{p}) = (g_1,...,g_k)$. Thus, for general $I$, I don't think that restriction to principal ideals actually simplifies the problem.
