Units in Near-rings

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.