Document Type : Original Manuscript


1 Department of Mathematics, Vali-e-Asr University of Rafsanjan , P.O.Box 7718897111, Rafsanjan, Iran.

2 Department of Mathematics, Shahid Bahonar University of Kerman, P.O.Box 76169133, Kerman, Iran.


In this paper, we give a generalization of the integral dependence from rings to modules. We study the stability of the integral closure with respect to various module theoretic constructions. Moreover, we introduce the notion of integral extension of a module and prove the Lying over, Going up and Going down theorems for modules.