Galois groups of modules and inverse polynomial modules

摘要

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 $M$, 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 $Bbb N$ (the set of all natural numbers).