Prime ideals in coordinate rings 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?
 A: 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. 
