  In this paper we introduce the semi-graded rings, which extend graded rings
and skew PBW extensions. For this new type of non-commutative rings we will
discuss some basic problems of non-commutative algebraic geometry. In
particular, we will prove some elementary properties of the generalized Hilbert
series, Hilbert polynomial and Gelfand-Kirillov dimension. We will extended the
notion of non-commutative projective scheme to the case of semi-graded rings
and we generalize the Serre-Artin-Zhang-Verevkin theorem. Some examples are
included at the end of the paper.
