Gröbner Bases for Ideals of σ-PBWExtensions

In this article, we introduce the σ-PWB extensions and construct the theory of Grobner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincare–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Grobner bases of left ideals.