Unit groups of compatible nearrings and Linz wreath products

Abstract

This is the capstone of a series of papers studying the units of a compatible nearring $R$ satisfying the descending chain condition on right ideals using a faithful compatible module $G$ of $R$. The approach is an inductive type one. Letting $H$ be a direct sum of isomorphic minimal $R$-ideals of $G$, assume we know the unit group of $R/Ann_R(G/H)$. The unit group of $R$ is then the extension group of the unit group of $1 + Ann_R(G/H)$ by the unit group of $R/Ann_R(G/H)$. Previous papers in this series have been devoted to the determination of the unit group of $1 + Ann_R(G/H)$ which has now been completed. This leaves the study of this extension group as the final step to complete, and that is the purpose of this paper. In its most general form, we will see that our extension group is a more general type of a wreath product we shall call a Linz wreath product in honor of its place of birth with an amalgamation. We then proceed to doing a number of examples where we explicitly determine these unit groups.