Almost Cohen-Macaulay modules
Journal
Communications in Algebra
Journal Volume
29
Journal Issue
2
Pages
781-787
Date Issued
2001
Author(s)
Abstract
Abstract A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R P , R P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X 1, …, X n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note. Keywords: Cohen–Macaulay ringsAlmost Cohen–Macaulay modules Acknowledgments
Type
journal article