  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.
