Krull dimension

Krull dimension of an integral domain

Let be an integral domain. Then the Krull dimension iff is a field, and iff every nonzero prime ideal is maximal.¹ ring

Proof

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

Note for every nonzero prime ideal is maximal by vacuity. Given , then any nonzero prime ideal is contained within a maximal ideal which is also prime (since Every ideal in a commutative ring is contained in a maximal ideal), but this must be equal to or else implies .

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Footnotes

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