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.