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 \( R \).
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The Krull dimension of a commutative ring is the supremum of the lengths of strict chains of prime ideals . The length counts inclusions, not the number of ideals.
A field has dimension zero. The integers have dimension one, with chains , and over a field has dimension . In algebraic geometry, chains of primes correspond to nested irreducible closed subsets, so this algebraic definition captures geometric dimension. It is distinct from the dimension of a vector space.