Hypernear-rings with a defect of distributivity

Since in a near-ring the distributivity holds just on one side (left or right), it seems naturally to study the behaviour and properties of the set of elements that ?correct? the lack of distributivity, in other words that elements that assure the validity of the distributivity. The normal subgroup of the additive structure of a near-ring generated by these elements is called a defect of distributivity of the near-ring. The purpose of this note is to initiate the study of the hypernear-rings (generalizations of near-rings, having the additive part a quasicanonical hypergroup) with a defect of distributivity, making a comparison with similar properties known for near-rings.