 As of 2017, it is in general unknown, whether curves in three-dimensional space are set-theoretic complete intersections.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"1, ..., 
"a""n" ∈ "m" such that all "a""i" 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.