Ore extensions of abelian groups with operators

Given a set A and an abelian group B with operators in A, in the sense of Krull and Noether, we introduce the Ore group extension $B[x;{\sigma }_B,{\delta }_B]$ as the additive group $B\left[x\right]$, with $A\left[x\right]$ as a set of operators. Here, the action of $A\left[x\right]$ on $B\left[x\right]$ is defined by mimicking the multiplication used in the classical case where A and B are the same ring. We derive generalizations of Vandermonde's and Leibniz's identities for this construction, and they are then used to establish associativity criteria. Additionally, we prove a version of Hilbert's basis theorem for this structure, under the assumption that the action of A on B is what we call weakly s-unital. Finally, we apply these results to the case where B is a left module over a ring A, and specifically to the case where A and B coincide with a non-associative ring which is left distributive but not necessarily right distributive.