  Planar nearrings play an important role in nearring theory, both from the
structural side as being close to generalised nearfields, as well as from an
applications perspective, in geometry and designs. 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 generalises 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.
