Extended modules and Ore extensions

Abstract

In this paper we investigate extended modules for a special class of Ore extensions. We will assume that $R$ is a ring and $A$ will denote the Ore extension $A:=R[x_1,\dots ,x_n;\sigma ]$ for which $\sigma$ is an automorphism of $R$, $x_i{x}_j=x_j{x}_i$ and $x_{i}r=\sigma (r)x_i$, for every $1\le i,j\le n$. With some extra conditions over the ring $R$, we will prove Vaserstein's, Quillen's patching, Horrocks' and Quillen-Suslin's theorems for this type of non-commutative rings.