In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains of prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) 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.
If P is a prime ideal of B and p its intersection with A, then The dimension formula for universally catenary rings says that equality holds if A is universally catenary.
means the transcendence degree (of quotient fields).
[1] Almost all Noetherian rings that appear in algebraic geometry are universally catenary.
The first example was found by Masayoshi Nagata (1956, 1962, page 203 example 2), who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary.
However A is not universally catenary, because if it were then the ideal mB of B would have the same height as mB∩A by the dimension formula for universally catenary rings, but the latter ideal has height equal to dim(A)=2.