  We prove that if R is a left Noetherian and left regular ring then the same
is true for any bijective skew PBW extension A of R. From this we get Serre's
Theorem for such extensions. We show that skew PBW extensions and its
localizations include a wide variety of rings and algebras of interest for
modern mathematical physics such as PBW extensions, well known classes of Ore
algebras, operator algebras, diffusion algebras, quantum algebras, quadratic
algebras in 3-variables, skew quantum polynomials, among many others. We
estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups.
