On N-groups and additive groups of near-rings with ATM (Q914812)

A right near-ring N is said to have ATM (almost trivial multiplication) if it is not a 0-near-ring and in N \(ab\in <a>\cap <b>\), where \(<x>\) is the additive group generated by \(x\in N\). If such a near-ring is \(\nu\)- primitive then it is simple, monogenic and has a right identity; moreover it is \(\nu\)-primitive qua N-group. A dihedral group cannot be the additive group of a near-ring with ATM. Finite abelian groups that can be the additive group of a near-ring with ATM are completely characterized in terms of invariants.