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Vietnam Journal of Mathematics 36:2(2008) 373-386

Some Homological Properties of Artinian Modules

Amir Mafi

Abstract. In this paper we show that if $(R,\fm)$ is a commutative Gorenstein local ring with maximal ideal $\fm$ and $M$ is an Artinian $R$-module, then $\depth (R) = \width(M) + \sup\{i \in \Bbb N_0: \, \Ext_R^i (E(R/ \fm), M) \neq 0\}$. Also, we prove that the following statements are equivalent:

(1) $R$ is Gorenstein.

(2) $R$ is Cohen-Macaulay and for any Artinian module $M$, $\text{rm fd}(E(M))\leq \text{rm fd}(M)$, where $E(M)$ is an injective envelope of $M$.

(3) $R$ is Cohen-Macaulay and for any finite length module $M$ of finite injective dimension, $\text{\rm id}(F(M))=\text{\rm id}(M)$, where $F(M)$ is a flat cover of $M$.

2000 Mathematics Subject Classification: 13D01, 13D05, 13D45, 13C11, 13C15, 13H05, 13H10.

Keywords: Artinian modules, Gorenstein injective, Local cohomology modules, Gorenstein rings, Depth.