Let be a ring.
The Krull dimension of is defined to be the supremum of the number of strict inclusions of prime ideals in .
The Krull dimension of a field is .
An integral domain is a field iff its Krull dimension is zero.
The Krull dimension of a PID which is not a field is .
The Krull dimension of for a field is .
The Krull dimension of for a noetherian ring of Krull dimension is . This is not always true if is not noetherian.