TITLE: Ideals of $\mathbb{Z}[X]$
QUESTION [51 upvotes]: Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ideals are well-known, but I'm interested in all the ideals. I couldn't find any literature about this. I don't even know the number of generators we need (Edit: Hurkyl has pointed out that any number appears!).
An ideal of $\mathbb{Z}[X]$ restricts to some ideal of $\mathbb{Z}$, say $n \mathbb{Z}$. For $n>0$ the ideal corresponds to some ideal in $\mathbb{Z}/n [X]$, and by CRT we may assume that $n$ is a prime power, say $n=p^k$. For $k=1$ we have the PID $\mathbb{F}_p[X]$, whose ideal structure is well-understood. What happens for $k=2$? The case $n=0$ seems to be even more complicated.
Since $\mathbb{Z}[X]$ is noetherian, every ideal is an intersection of primary ideals, and we should classify them first? What about other special classes of ideals, for example radical ones?
If the question is too naive in this generality, what about special cases, or slight weakenings? For example, is it possible to classify all (reduced, finite, coprimary, ...) commutative rings which are generated by a single element? This comes down to the classification of the ideals of $\mathbb{Z}[X]$ modulo isomorphic quotient rings.

REPLY [8 votes]: There are generalizations of Grobner/standard ideal basis algorithms that work over coefficient rings more general than fields, e.g. certain PIDs (which, in particular, apply to $\rm\,\Bbb Z[x]).$ There is, by now, much literature on such topics. Perhaps the easiest way to chase it down is to start with the references listed in Buchberger's survey [1] on the history of the Grobner basis algorithm.
See in particular the work mentioned in section "Generalizations of the CPC approach for polynomial rings", p. 19. If you chase citations of the work mentioned there, it will quickly lead you to the most recent work on said generalizations.
[1] B. Buchberger. History and basic features of the critical pair / completion procedure.
Journal of Symbolic Computation, 3(1/2) (1987) 3-38