Matrix maps over planar near-rings

Abstract

Following a method by Meldrum and van der Walt, near-rings of matrix maps are defined for general near-rings, not necessarily with identity. The influence of one-sided identities is discussed. When the base near-ring is integral and planar, the near-ring of matrix maps is shown to be simple. Various types of primitivity of the near-ring of matrix maps are discussed when the base near-ring is planar but not integral. Finally, an open problem concerning bijective matrix maps is solved.

Distribution and Generalized Center in Planar Nearrings

“…It is worth noting in passing that nearvector spaces and the homogeneous mappings of them to themselves are closely related to questions about nearring matrices over the associated nearfield. Thus we hope that future work here could shed light on the question raised in the final section of [14] as to the inverses of units in matrix nearrings over planar nearfields.…”

“…Nearrings are the nonlinear generalization of rings, having only one distributive law. Planar nearrings are a special class of nearrings that generalize nearfields, themselves a generalization of fields, which play an important role in the structural theory of nearrings [14,20], as well as having important geometric and combinatorial properties [6,10,15].…”