In mathematics, a commutative ring R is catenary if for any pair of prime ideals
any two strictly increasing chains
are contained in maximal strictly increasing chains from p to q of the same length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions.
A ring is called universally catenary if all finitely generated algebras over it are catenary rings.