Distribution and Generalized Center in Planar Nearrings

Abstract

Nearrings are the nonlinear generalization of rings. Planar nearrings play an important role in nearring theory, both from the structural side, being close to generalized nearfields, as well as from an applications perspective, in geometry and combinatorial designs related to difference families. In this paper we investigate the distributive elements of planar nearrings. If a planar nearring has nonzero distributive elements, then it is an extension of its zero multiplier part by an abelian group. In the case that there are distributive elements that are not zero multipliers, then this extension splits, giving an explicit description of the nearring. This generalizes the structure of planar rings. We provide a family of examples where this does not occur, the distributive elements being precisely the zero multipliers. We apply this knowledge to the question of determining the generalized center of planar nearrings as well as finding new proofs of other older results.