Some properties of Artinian modules and applications

Let (R, m) be a Noetherian local ring and A be an Artinian R-module. Consider the following property for A: Ann(R)(0 : (A) p) = p for all p is an element of Var(Ann(R)(A)). (*) In this paper, we study the property (*) of A in order to investigate the relation of system of parameters between A and the ring R/Ann(R) A. We also show that the property (*) of A 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.