Ore Extensions of Abelian Groups with Operators

TL;DR

This paper introduces Ore group extensions for abelian groups with operators, generalizing classical identities and establishing a Hilbert basis theorem under specific conditions, with applications to modules and non-associative rings.

Contribution

It develops a new construction of Ore extensions for abelian groups with operators, generalizes key identities, and proves a Hilbert basis theorem in this context.

Findings

Derived generalized Vandermonde and Leibniz identities.

Established associativity criteria for the extension.

Proved a version of Hilbert's basis theorem under weakly s-unital action.

Abstract

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 $B[x]$, with $A[x]$ as a set of operators. Here, the action of $A[x]$ on $B[x]$ 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.