Definition for "Krull Dimension" in the Quantum Dictionary

- Natural & Formal Sciences -

Mathematics & Logic Dictionary

The Krull dimension of a commutative ring is the supremum of the lengths n of chains of prime ideals P0 ⊂ P1 ⊂ ... ⊂ Pn; it provides an algebraic measure of 'dimension' that often corresponds to the geometric dimension of Spec of the ring.

- Mathematics & Logic -

Pure Mathematics Dictionary

The supremum of lengths n of chains of prime ideals P0 ⊂ P1 ⊂ ... ⊂ Pn in a commutative ring (or of irreducible closed subsets in a topological scheme), providing an algebraic measure of geometric dimension.

- Pure Mathematics -

Algebra Dictionary

The supremum of the lengths n of chains of prime ideals (or prime subobjects) P0 ⊂ P1 ⊂ ... ⊂ Pn in a commutative ring or analogous algebraic structure; a primary algebraic notion of dimension.