Maximal Ideals of Different Heights

examples-counterexamples ring-theory commutative-algebra maximal-and-prime-ideals

I want to see how it is possible for rings to have maximal ideals of different heights. For this, I need to see various cases of such rings. I can construct one case by localization using methods outlined in this math SE question.

Now I want to construct other cases with the following property:
The ring $R$ has infinitely many maximal ideals, one with height $3$ and one with height $2$. Any other case of rings with maximal ideals of different heights are welcomed.


Take a ring $R_1$ which has a maximal ideal of height 2, a ring $R_2$ with a maximal ideal of height 3, and form $R=R_1\times R_2\times \mathbb Z$.

That would suit your requirements.

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