Hilbert's Basis Theorem for generalized nonassociative Ore extensions

We introduce a broader class of nonassociative Ore extensions that unifies and generalizes several earlier constructions. We prove generalizations of Hilbert's Basis Theorem for this class, showing that they arise immediately from the existence of Euclidean division algorithms. These results extend Hilbert's Basis Theorem to new families of nonassociative, noncommutative polynomial rings and establish a novel and direct connection between Euclidean division algorithms and the left and right Noetherianity of such rings.