On local near-rings with Miller–Moreno multiplicative group

Abstract

A near-ring R with identity is local if the set L of all its noninvertible elements is a subgroup of the additive group R +. We study local near-rings of order 2 n whose multiplicative group R * is a Miller–Moreno group, i.e., a non-abelian group all proper subgroups of which are abelian. In particular, it is proved that if L is a subgroup of index 2 m in R +, then either m is a prime number for which 2 m − 1 is a Mersenne prime or m = 1. In the first case, n = 2m, the subgroup L is elementary abelian, the exponent of R + does not exceed 4; and R * is of order 2 m (2 m − 1)). In the second case, either n < 7 or the subgroup L is abelian and R * is a nonmetacyclic group of order 2 n−1 whose exponent does not exceed 2 n−4.