Bulletin of the
Korean Mathematical Society BKMS

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).

Keywords: injective module, injective envelope, Galois group, inverse polynomial module

MSC numbers: Primary 16E30; Secondary 13C11, 16D80