Timothy Boykett,
"Distribution and generalized centre in planar nearrings"
, in Acta Scientiarum Mathematicarum (Szeged), Vol. 86, Nummer 1-2, Seite(n) 11-29, 2020, ISSN: 2064-8316
Original Titel:
Distribution and generalized centre in planar nearrings
Sprache des Titels:
Englisch
Original Kurzfassung:
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 an abelian group by
its zero multiplier part. In the case that there are distributive elements that are
not zero multipliers, then this extension splits, giving an explicit description
of the nearring, a coordinatisation result. 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 centre of planar
nearrings as well as finding new proofs of older results.