Editing Commutative ring (section)

=== Cohen–Macaulay rings ===
The [[depth (ring theory)|depth]] of a local ring ''R'' is the number of elements in some (or, as can be shown, any) maximal regular sequence, i.e., a sequence ''a''<sub>1</sub>, ..., 
''a''<sub>''n''</sub> &isin; ''m'' such that all ''a''<sub>''i''</sub> are non-zero divisors in
{{block indent|1= ''R'' / (''a''<sub>1</sub>, ..., ''a''<sub>''i''&minus;1</sub>). }}
For any local Noetherian ring, the inequality
{{block indent|1= depth(''R'') &le; dim(''R'') }}
holds. A local ring in which equality takes place is called a [[Cohen–Macaulay ring]]. Local complete intersection rings, and a fortiori, regular local rings are Cohen–Macaulay, but not conversely. Cohen–Macaulay combine desirable properties of regular rings (such as the property of being [[universally catenary ring]]s, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.{{sfn|Eisenbud|1995|loc=Corollary 18.10, Proposition 18.13|ps=}}