1. Ore Extensions of Abelian Groups with Operators
- Author
-
Bäck, Per, Lundström, Patrik, Öinert, Johan, and Richter, Johan
- Subjects
Mathematics - Rings and Algebras ,Mathematics - Representation Theory ,16S36, 16W22, 16W70, 17A99, 17D99, 20K27 - Abstract
Given a set $A$ and an abelian group $B$ with operators in $A$, we introduce the Ore group extension $B[x ; \delta_B , \sigma_B]$ as the additive group $B[x]$, with $A[x]$ as a set of operators, the action of $A[x]$ on $B[x]$ being 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.
- Published
- 2024