Hilbert's basis theorem for non-associative and hom-associative Ore extensions

We prove a hom-associative version of Hilbert's basis theorem, which includes as special cases both a non-associative version and the classical associative Hilbert's basis theorem for Ore extensions. Along the way, we develop hom-module theory. We conclude with some examples of both non-associative and hom-associative Ore extensions which are all noetherian by our theorem.
Comment: 18 pages; simplified the proof of Proposition 12; changed the title, abstract and introduction, added an example and a corollary together with some minor changes; corrected some minor typos in reference list; major revision including new title, shorter proofs and four new examples; added an example and updated an example, adjusted the language; added/adjusted examples; minor update