Exceptional units in the residue class rings of global fields

Su Hu, Min Sha

Published 2016-12-14Version 1

Let $R$ be a commutative ring with $1\in R$ and $R^{*}$ its group of units. A unit $u\in R^*$ is called exceptional if $1-u\in R^{*}$. In the case $R=\mathbb{Z}/n\mathbb{Z}$ of residue classes modulo $n$, recently Sander has determined the number of representations of an arbitrary element $c\in \mathbb{Z}/n\mathbb{Z}$ as the sum of two exceptional units. This result has been immediately generalized by Yang and Zhao, which gives an exact formula for the number of ways to represent each element of $\mathbb{Z}/n\mathbb{Z}$ as the sum of $k \ge 2$ exceptional units. In this paper, we show that the method of Yang and Zhao in fact leads to a generalization in any residue class ring of global fields. Moreover, we completely determine the additive structure of such exceptional units.