Motivated by Smith’s work (Trans Am Math Soc 356(7):2927–2944, 2003; Trans Am Math Soc 368(11):8295–8302, 2016) on maps between non-commutative projective spaces of the form \(\textrm{Proj}_{nc}\ A\) in the setting of non-commutative projective geometry developed by Rosenberg and Van den Bergh, and the notion of schematicness introduced by Van Oystaeyen and Willaert (J Pure Appl Algebra 104(1–3):109–122, 1995) to \({\mathbb {N}}\) -graded rings with the aim of formulating a non-commutative scheme theory à la Grothendieck and Dieudonné (Éléments de Géométrie Algébrique II, vol 8, 1961), in this paper we consider a first approach to maps in the Smith’s sense in the more general setting of non-commutative projective spaces over semi-graded rings defined by Lezama and Latorre (J Algebra Comput 27(4):361–389, 2017) . We extend Smith’s key result (Smith et al. 2003, Theorem 3.2; Smith 2016, Theorem 1.2) from the category of schematic \({\mathbb {N}}\) -graded rings to the category of schematic semi-graded rings.
