Local Noetherian domain implies equidimensional

Statement
Any local Noetherian domain is equidimensional.

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
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.