stylistic:style_transfer, structural:expansion, structural:summarization
Cohen–Macaulay rings

The 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,..., a ∈ m such that all a are non-zero divisors in

For any local Noetherian ring, the inequality

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 rings, which means that the (co)dimension of primes is well-behaved), but are also more robust under taking quotients than regular local rings.