Local Noetherian domain implies equidimensional

Statement

Any local Noetherian domain is equidimensional.

Definitions used

Local Noetherian domain

Further information: local Noetherian domain

A Noetherian ring is termed a local Noetherian domain if:

• 0 is a prime ideal (and hence the unique minimal prime ideal)
• There is a unique maximal ideal

Equidimensional ring

Further information: Equidimensional ring

A Noetherian ring is termed equidimensional if:

• All its minimal prime ideals have the same dimension
• All its maximal ideals have the same codimension

Proof

The proof is by definition: in a local Noetherian domain, there is a unique maximal ideal, and a unique minimal prime ideal. Hence, by definition, all maximal ideals have the same codimension, and all minimal primes have the same dimension, so the ring is equidimensional.