A commutative ring in which there exist two prime ideals p ⊂ q for which saturated chains of prime ideals between p and q have different finite lengths; equivalently, the ring fails the catenary property that requires all saturated chains between two primes to have the same length.

Principle

Catenarity demands uniformity of heights: between any two comparable primes the lengths of maximal chains should be independent of the chosen chain. A Noncatenary Ring violates this uniformity, so height behaves irregularly.

Demonstration

There are explicit examples of Noetherian integral domains constructed to be noncatenary; in such a ring one can find primes p ⊂ q and two maximal chains p = p0 ⊂ p1 ⊂ ... ⊂ pr = q and p = q0 ⊂ q1 ⊂ ... ⊂ qs = q with r ≠ s, demonstrating unequal lengths between the same endpoints.

Misapplication

Assuming dimension-theoretic formulas that rely on catenarity (for instance equating dimension differences with heights computed along arbitrary chains), or using geometric intuition that dimensions of localizations vary predictably, can lead to false conclusions in noncatenary contexts.

Consequence

Dimension theory becomes more subtle: computations of relative heights, equidimensionality assertions, and many dimension-sensitive arguments require checking chain-independence; properties like equidimensionality and certain dimension formulas may fail or require extra hypotheses.

Reversal

A catenary ring, where for any pair of comparable prime ideals every saturated chain between them has the same length, simplifying dimension theory and height computations.

Boundary

This notion applies to chains of prime ideals in commutative rings and is a purely prime-ideal/dimension-theoretic condition; it does not directly assert Noetherianness, regularity, or other finiteness conditions, although those interact. Noncatenarity refers to chain-length nonuniformity and excludes unrelated pathologies of modules or radicals.

Semantic Tension

There is tension between catenarity and related notions such as equidimensionality or Cohen–Macaulayness; a ring can be equidimensional yet fail catenarity, and catenarity is independent of some finiteness conditions, so different intuitions about ‘well-behaved dimension’ compete.

Synthesis

A Noncatenary Ring is one in which the expected uniformity of prime-ideal chain lengths breaks down: between the same primes one can find maximal chains of distinct lengths, forcing careful local checks in dimension-theoretic arguments rather than relying on a global chain-length invariance.