We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew P BW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew P BW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as P BW 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.

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We study Zariski cancellation problem for noncommutative algebras that are not necessarily domains.2000 Mathematics Subject Classification. Primary 16P99, 16W99.

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 noncommutative 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.

Las extensiones PBW torcidas graduadas fueron definidas por el primer autor como una generalización de las extensiones de Ore iteradas graduadas [36]. El propósito de este artículo es estudiar las condiciones Artin-Schelter regular y Calabi-Yau (torcida) para esta clase de extensiones. Demostramos que cada extensión PBW torcida cuasi-conmutativa graduada de un álgebra Artin-Schelter regular también es Artin-Schelter regular, y, como consecuencia, que cada extensión PBW torcida cuasi-conmutativa graduada de un álgebra conexa Calabi-Yau torcida es Calabi-Yau torcida. Finalmente, mostramos que las extensiones PBW torcidas graduadas de álgebras Auslander-regular finitamente presentadas y conexas son Calabi-Yau torcidas.