 ##  [Krull Dimension](/krull-dimension) 

  ##  [Krull Dimension](https://mathlogic.quantumdictionary.io/krull-dimension-0) 

  
 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural &amp; Formal Sciences -

**Mathematics &amp; Logic Dictionary**

 
Definition

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.

 
 ##  [Krull Dimension](https://puremath.quantumdictionary.io/krull-dimension-1) 

  
 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics &amp; Logic -

**Pure Mathematics Dictionary**

 
Definition

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.

 
 ##  [Krull Dimension](https://algebra.quantumdictionary.io/krull-dimension-2) 

  
 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 
Definition

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.