Some properties of Artinian modules and applications

Abstract

Let [Formula: see text] be a Noetherian local ring and [Formula: see text] be an Artinian [Formula: see text]-module. Consider the following property for [Formula: see text] : [Formula: see text] In this paper, we study the property (∗) of [Formula: see text] in order to investigate the relation of system of parameters between [Formula: see text] and the ring [Formula: see text]. We also show that the property (∗) of [Formula: see text] has strong connection with the structure of base ring. Some applications to cofinite Artinian module are given. These are generalizations of [N. Abazari and K. Bahmanpour, A note on the Artinian cofinite modules, Comm. Algebra. 42 (2014) 1270–1275; G. Ghasemi, K. Bahmanpour and J. Azami, On the cofiniteness of Artinian local cohomology modules, J. Algebra Appl. 15(4) (2016), Article ID: 1650070, 8 pp.] A generalization of Lichtenbaum–Hartshorne Vanishing Theorem is also given in this paper.