On almost Cohen–Macaulayness of quotient modules

Abstract

Let $$(R,\mathfrak {m})$$ be a local ring and M a non-zero finitely generated R-module of dimension d. The main purpose of this paper is to provide a characterization of almost Cohen–Macaulay R-modules. After that, we show that if I is an ideal of R such that M / IM is an almost Cohen–Macaulay R-module of dimension d and there is a system of parameters $$x_1,\ldots ,x_d$$ for M, in $$\mathfrak {m}$$ , such that $$I\subseteq (x_1,\ldots ,x_d)$$ then either $$I\subseteq {\text {Ann}}_R(M)$$ or there exists an index $$1\le j\le d$$ such that $$x_j\in Z_R(M/(I+(x_1,\ldots ,{\widehat{x_j}},\ldots ,x_d))M)$$ and $$I\nsubseteq (x_1,\ldots ,{\widehat{x_j}},\ldots ,x_d)$$ .