Minimal left ideals of near-rings

We show that a finite minimal left ideal L of a zero symmetric near-ring N is a planar near-ring if L is not contained in the radical J2(N). This result will follow from a more general discussion on minimal N-subgroups of a near-ring. Then we discuss some consequences of this result when applied to the structure theory of near-rings. Finally we transfer our results to rings and deal with some ring theoretic questions concerning “trivial” multiplications in rings.