Ore Extensions of Abelian Groups with Operators
Per B\"ack, Patrik Lundstr\"om, Johan \"Oinert, and Johan Richter

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 and an abelian group with operators in , in the sense of Krull and Noether, we introduce the Ore group extension as the additive group , with as a set of operators. Here, the action of on is defined by mimicking the multiplication used in the classical case where and 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 on is what we call weakly -unital. Finally, we apply these results to the case where is a left module over a ring , and specifically to the case where and coincide with a non-associative ring which is left distributive but not…
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Taxonomy
Topicsadvanced mathematical theories
