It is well-known that in $\mathbb{Z}[X]$ we do have non-principal ideals, for example $(2,x)$. This is an ideal with two generators. Now I was wondering if there exists an ideal with three generators, which cannot be generated by two elements. (And of course, if so, if we can find ideals with $n$ generators which cannot be generated by $n-1$ elements).
I do have one suggestion: $(8, 4x, 2x^2)$, found by trial-and-error.
My question is twofold:
- Is this an ideal as described above?
- Is there a more constructive way to think about this question?
From the way the example is set the ideal $(2^n,2^{n-1}x,2^{n-2}x^2,...,2x^{n-1},x^n)$ is an ideal with $n$ honest generators in $\Bbb Z[x]$ because we cannot generate any of these generators by using the preceding ones.