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In this paper we define the algebraic sets and the ideal of points forbijective skew PBW extensions with coefficients in left Noetherian domains.Some properties of affine algebraic sets of commutative algebraic geometry willbe extended, in particular, a Zariski topology will be constructed. Assumingadditionally that the extension is quasi-commutative with polynomial center andthe ring of coefficients is an algebraically closed field, we will prove anadapted version of the Hilbert's Nullstellensatz theorem that covers theclassical one. The Gr\"obner bases of skew PBW extensions will be used fordefining the algebraic sets and for proving the main theorem. Many key algebrasand rings coming from mathematical physics and non-commutative algebraicgeometry are skew PBW extensions.

Algebraic sets, ideals of points and the Hilbert's Nullstellensatz theorem for skew PBW extensions