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.
For a Noetherian local ring R, the depth of the zero module is ∞,[2] whereas the depth of a nonzero finitely generated R-module M is at most the Krull dimension of M (also called the dimension of the support of M).
[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.