In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a commutative ring

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

 R is catenary if for any pair of prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

s

p, q,

any two strictly increasing chains

p=p₀ ⊂p₁ ... ⊂p_n= q of prime ideals

are contained in maximal strictly increasing chains from p to q of the same (finite) length. In other words, there is a well-defined function from pairs of prime ideals to natural numbers, attaching to p and q the length of any such maximal chain. In a geometric situation, in which the dimension of an algebraic variety

Dimension of an algebraic variety

In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...

 attached to a prime ideal will decrease as the prime ideal becomes bigger, there is reason to believe that the length of such a chain will conform to n = difference in dimensions, with dimension decrementing by 1 at each step.

A ring is called universally catenary if all finitely generated rings over it are catenary.

The word 'catenary' is derived from the Latin word catena, which means "chain".

Examples

Almost all Noetherian ring

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

s that appear in algebraic geometry are universally catenary.
In particular the following rings are universally catenary:

• Complete Noetherian local ring
  Local ring
  In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
  
  s
• Dedekind domain
  Dedekind domain
  In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
  
  s (and fields)
• Cohen-Macaulay ring
  Cohen-Macaulay ring
  In mathematics, a Cohen–Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality....
  
  s
• Any localization
  Localization of a ring
  In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
  
   of a universally catenary ring
• Any finitely generated algebra over a universally catenary ring.

It is very hard to construct examples of rings that are not universally catenary. The first example was found by Masayoshi Nagata

Masayoshi Nagata

Masayoshi Nagata was a Japanese mathematician, known for his work in the field of commutative algebra....

in 1956 (see Nagata (1962) page 203 example 2), who produced a Noetherian local domain that was catenary but not universally catenary.