In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''R'' is catenary if for any pair of
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s
:''p'', ''q'',
any two strictly increasing chains
:''p''=''p''
0 ⊂''p''
1 ... ⊂''p''
''n''= ''q'' 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
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.
Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutati ...
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.
The word 'catenary' is derived from the Latin word ''catena'', which means "chain".
There is the following chain of inclusions.
Dimension formula
Suppose that ''A'' is a Noetherian domain and ''B'' is a domain containing ''A'' that is finitely generated over ''A''. 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. Here κ(''P'') is the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
of ''P'' and tr.deg. means the transcendence degree (of quotient fields). In fact, when ''A'' is not universally catenary, but