Graded extensions in K[Q, sigma]

TL;DR

This paper classifies and analyzes graded extensions of a valuation ring within skew group rings formed over a division ring K with rational group Q and automorphisms, revealing their structural types.

Contribution

It provides a classification of graded extensions of valuation rings in skew group rings over division rings, detailing their structural types.

Findings

Classified graded extensions into two types

Analyzed the structure of these extensions

Provided a framework for understanding extensions in skew group rings

Abstract

Let V be a total valuation ring of a division ring K, Q be the additive group of the rational numbers, Aut(K) be the group of automorphisms of K. Let sigma be a group homomorphism from Q to Aut(K). Let K[Q, sigma ] be the skew group ring of Q over K. In this paper, we classify graded extensions of V in K[Q, sigma] into two types and study the structure of them.