# If all prime ideals are maximal, then why is krull dimension 0?

I know that if all prime ideals are maximal ideals (like in PID or Artin ring) then why is their krull dimension $$0$$? Let's say we have $$p$$ a prime ideal in Artin ring, then it is maximal so we have $$(0)\subsetneq p$$ where strict chain of prime ideals $$(0)$$ and $$p$$ (which is maximal therefore prime) would give krull dimension of $$1$$ not $$0$$ for Artin ring.

Also, for a PID that is not a field, the krull dimension is $$1$$. Again all nonzero prime ideals are maximal in PID therefore for $$(0)\subsetneq p$$, where chain of prime ideals $$(0)$$ and $$p$$ would give it dimension $$1$$ as expected.

So is the logic for both PID that's not a field and the Artin ring same for dimension? Then their dimensions should be $$1$$. I am slightly confused with the definition. Please explain.

Looks like you've fallen into two misapprehensions:

so we have $$(0)\subsetneq p$$ where $$p$$ is maximal. Then the krull dimension should be $$1$$ not $$0$$.

You did not exhibit a chain of prime ideals. In a right Artinian ring, the zero ideal is only prime if the ring is already a simple ring, and so there aren't any other proper ideals to be had. In particular for a commutative Artinian ring, the zero ideal is prime only if the ring is already a field.

Also, for a PID that is not a field, the krull dimension is $$1$$. Again all prime ideals are maximal in PID, so $$(0)⊊𝑝$$ would give it dimension 1 as expected.

All nonzero prime ideals are maximal in a PID, but the ideal $$\{0\}$$ is prime and not maximal, so a nonfield PID does not satisfy what is required. (What was required was all primes maximal.)

Proving the statement is straightforward enough: suppose you have a prime ideal $$P$$ that isn't maximal. Pick any maximal ideal $$M$$ containing $$P$$. $$M$$ is also prime. Then $$P\subsetneq M$$ witnesses that the Krull dimension is at least $$1$$. Using the contrapositive, primes are maximal in a zero-dimensional ring.

• It makes sense now. Thanks rschwieb.
– John
Commented May 9, 2023 at 11:18
• @John :tips-hat: Commented May 9, 2023 at 14:16