On the summands of near-rings with ATM (Q1072626)

A near-ring N has an almost trivial multiplication (ATM) if the product of two elements belongs to the intersection of the additive cyclic groups generated by these two elements. In a near-ring with ATM all the subgroups are invariant subnear-rings. We show that every finite near-ring with ATM can be decomposed into a direct sum where the summands are either near-rings defined on cyclic groups or near-rings whose minimal ideals are zero near-rings. By using some elementary number theory we show how to construct these summands on cyclic groups.