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

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.