Are maximal ideals and principal ideals independent?

Question

How can I show that the properties maximal and principal can be true independently for an ideal?

I can find one example of each:

1. A maximal principal ideal
2. A maximal non-principal ideal
3. A non-maximal principal ideal
4. A non-maximal non-principal ideal

Is there a shortcut?

Answer 1

I think the easiest examples come from algebraic geometry.

An example of a principal ideal that is maximal is given by $\langle x\rangle \subset k[x]$, which corresponds to the zero point in $\mathbb{A}_k$ and thus is maximal.

An example of a principal non maximal ideal is given by $\langle 0\rangle\subset \mathbb{Z}$.

For a maximal non principal ideal take $\langle x,y\rangle \subset k[x,y]$.

For something that is neither maximal nor principal, take the vanishing of two polynomials which does not consist of only one point.