In this survey paper we first present the main properties of sequentially Cohen-Macaulay modules. Some basic examples are provided to help reader with quickly getting acquainted topic. We then discuss two generalizations notion sequential Cohen-Macaulayness which inspired by a theorem J\"urgen Herzog and third author.

In the late 1960s Auslander and Bridger [2] introduced a notion of approximation which they used to prove that every module whose n syzygy is n-torsionfree can be described as the quotient of an n-spherical module by a submodule of projective dimension less than n. About two decades later Auslander and Buchweitz [3] introduced the notion of Cohen-Macaulay approximation which they used to show t...

Let [Formula: see text] where be a complete regular local ring of dimension text], for some and an MCM text]-module with then we prove that text]. If is hypersurface infinite residue field let or Our paper the first systematic study depth associated graded modules over rings.