Units in Near-rings

TL;DR

This paper explores properties of near-rings related to units, generalizing concepts from nearfields, and investigates their structural characteristics, constructions, and examples to understand their algebraic behavior.

Contribution

It introduces and analyzes two related properties of units in near-rings, providing new constructions and examples that distinguish these properties from classical ring theory.

Findings

Units with zero form an additive subgroup in certain near-rings.

Units act without fixed points on the additive structure.

Proper examples are neither simple nor J2-semisimple.

Abstract

We investigate near-ring properties that generalize nearfield properties about units. We study zero symmetric near-rings $N$ with identity with two interrelated properties: the units with zero form an additive subgroup of $(N,+)$; the units act without fixedpoints on $(N,+)$. There are many similarities between these cases, but also many differences. Rings with these properties are fields, near-rings allow more possibilities, which are investigated. Descriptions of constructions are obtained and used to create examples showing the two properties are independent but related. Properties of the additive group as a $p$-group are determined and it is shown that proper examples are neither simple nor $J_2$-semisimple.