Given a commutative ring R and an R-module M, an element r in R is called a non-zero-divisor on M if r m = 0 implies m = 0 for m in M. An M-regular sequence is a sequence r1, ..., rd of elements of R such that r1 is a not a zero-divisor on M and ri is a not a zero-divisor on M/(r1, ..., ri−1)M for i = 2, ..., d. [1] Some authors also require that M/(r1, ..., rd)M is not zero.
Intuitively, to say that r1, ..., rd is an M-regular sequence means that these elements "cut M down" as much as possible, when we pass successively from M to M/(r1)M, to M/(r1, r2)M, and so on.
But if R is a Noetherian local ring and the elements ri are in the maximal ideal, or if R is a graded ring and the ri are homogeneous of positive degree, then any permutation of a regular sequence is a regular sequence.
That is, the depth of R is the maximum length of a regular sequence in the maximal ideal.
[3] An important case is when the depth of a local ring R is equal to its Krull dimension: R is then said to be Cohen-Macaulay.
Similarly, a finitely generated R-module M is said to be Cohen-Macaulay if its depth equals its dimension.
Similar examples can be found by looking at minimal generators for the ideals generated from reducible schemes with multiple components and taking the subscheme of a component, but fattened.