On the additive and multiplicative structures of the exceptional units in finite commutative rings

TL;DR

This paper investigates the structure of exceptional units in finite commutative rings, characterizing their additive and multiplicative properties and establishing conditions for the ring to be generated by these units.

Contribution

It provides a detailed analysis of the additive and multiplicative structures of exceptional units and characterizes when such units generate the entire ring.

Findings

Determined the structure of exceptional units in finite commutative rings.

Established a necessary and sufficient condition for the ring to be generated by exceptional units.

Provided insights into the algebraic properties of units in finite rings.

Abstract

Let $R$ be a commutative ring with identity. A unit $u$ of $R$ is called exceptional if $1-u$ is also a unit. When $R$ is a finite commutative ring, we determine the additive and multiplicative structures of its exceptional units; and then as an application we find a necessary and sufficient condition under which $R$ is generated by its exceptional units.