Galois groups of modules and inverse polynomial modules

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal(phi). Let R be a left noetherian ring and E be an injective envelope of All, then there is an injective envelope E[x(-1)] of an inverse polynomial module M[x(-1)] as a left R[x]-module and we can define an associative Galois group Gal(phi[x(-1)]). In this paper we describe the relations between Gal(phi) and Gal(phi[x(-1)]). Then we extend the Galois group of inverse polynomial module and can get Gal(phi[x(-s)]), where S is a submonoid of N (the set of all natural numbers).