The Krull dimension of a commutative ring is the supremum of the lengths of strict chains of prime ideals.
The height of an ideal is the infimum of the heights of prime ideals containing it:The height of a prime is the supremum of the lengths of strict prime chains ending at that prime.
For a Noetherian local ring , the Krull dimension of , the Krull dimension of the associated graded ring , and the order of the pole at of its Hilbert series are equal.
If is a non-zero-divisor in a finite-dimensional Noetherian ring , thenEvery prime chain above can be extended downward by a minimal prime of , because a non-zero-divisor belongs to no minimal prime.
Articles by others on the same topic
The Krull dimension is a concept in commutative algebra and algebraic geometry that measures the "size" or complexity of a ring or a space in terms of its prime ideals. More formally, the Krull dimension of a ring \( R \) is defined as the supremum of the lengths of all chains of prime ideals in \( R \).