Catenary ring

Definition
A commutative unital ring is termed a catenary ring or chain ring or is said to satisfy the Saturated Chain Condition if it is Noetherian and satisfies the following equivalent conditions:


 * If $$P < P_1 < P_2 < Q$$ is a strictly ascending chain of prime ideals, and $$P'$$ is a prime ideal between $$P$$ and $$Q$$, then there is either a prime ideal between $$P$$ and $$P'$$ or a prime ideal between $$P'$$ and $$Q$$
 * Given two prime ideals $$P$$ and $$Q$$ such that $$P \subset Q$$, the length of any saturated chain of primes between $$P$$ and $$Q$$ (i.e. a chain of primes in which no more primes can be inserted in between) is determined independent of the choice of chain

Note that being catenary does not guarantee that any two fully saturated chains of primes (i.e. any two chains of primes which are saturated and cannot be extended in either direction) have the same length. The problem is that the starting and ending points of the chains may differ: there may be many different maximal ideals and many different minimal prime ideals.