Krull dimension

Krull dimension of an integral domain

Let 𝑅 be an integral domain. Then the Krull dimension dim⁡𝑅 =0 iff 𝑅 is a field, and dim⁡𝑅 ≤1 iff every nonzero prime ideal is maximal.¹ ring

Proof

^C1. So if 𝑅 is a field, dim⁡𝑅 =0. Conversely, if dim⁡𝑅 =0 then 0 is a maximal ideal (as Every commutative ring has a maximal ideal and A maximal ideal in a commutative ring is prime), and thus 𝑅 has no nonzero proper ideals: 𝑅 is a field.

Note for dim⁡𝑅 =0 every nonzero prime ideal is maximal by vacuity. Given dim⁡𝑅 =1, then any nonzero prime ideal 𝔭 is contained within a maximal ideal 𝔪 which is also prime (since Every ideal in a commutative ring is contained in a maximal ideal), but this must be equal to 𝔭 or else 0 ⊊𝔭 ⊊𝔪 implies dim⁡𝑅 >1.

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Footnotes

1. 2022. Algebraic number theory course notes, ¶1.23, p. 15 ↩