Extended modules and Ore extensions

TL;DR

This paper extends classical module theorems to a special class of non-commutative Ore extensions, broadening the understanding of projective modules over these rings.

Contribution

It proves versions of Vaserstein's, Quillen's patching, Horrocks', and Quillen-Suslin's theorems for specific Ore extensions with automorphisms.

Findings

Classical module theorems are valid for certain non-commutative Ore extensions.

Conditions on the ring R allow these theorems to hold in the non-commutative setting.

The results extend the theory of projective modules to new classes of non-commutative rings.

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 $R$, $x_ix_j=x_jx_i$ and $x_ir=\sigma(r)x_i$, for every $1\leq i,j\leq n$. 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.