# Some open problems in the context of skew PBW extensions and semi-graded rings

## TL;DR

This paper explores open problems in non-commutative algebra and algebraic geometry using skew PBW extensions and semi-graded rings, focusing on conjectures, isomorphisms, and properties like noetherianity.

## Contribution

It analyzes key open problems and reformulates conjectures within the framework of skew PBW extensions and semi-graded rings, providing new insights and partial solutions.

## Key findings

- Analysis of isomorphisms related to the Gelfand-Kirillov conjecture
- Discussion of Serre's conjecture for specific skew PBW extensions
- Reformulation of noetherianity and Zariski cancellation questions for semi-graded rings

## Abstract

In this paper we discuss some open problems of non-commutative algebra and non-commutative algebraic geometry from the approach of skew $PBW$ extensions and semi-graded rings. More exactly, we will analyze the isomorphism arising in the investigation of the Gelfand-Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain. The Serre's conjecture will be discussed for a particular class of skew $PBW$ extensions. The questions about the noetherianity and the Zariski cancellation property of Artin-Schelter regular algebras will be reformulated for semi-graded rings. Advances for the solution of some of the problems are included.