Zerosymmetric idempotent near-rings with Abelian additive groups

The goal of this paper is to clarify a structure of the near-rings indicated in the title (shortly, ZPIR-near-rings). It is shown that any such near-ring $N$ is weakly commutative and poset $N$ endowed by the natural order relation as a reduced near-ring, is a union of Boolean lattices and may be presented as a coextension of the generalized Boolean lattice by the family of left bands. At the end of the article one defines an ideally hereditary radical in the class of all ZPIR-near-rings, the corresponding semisimple class consisting of Boolean rings.