On local near-rings with Miller?Moreno multiplicative group - Raievska, M. Yu.; Sysak, Ya. P. (Ukrains’kyi Matematychnyi Zhurnal, 2012)
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 the 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 for which 2^m - 1 is a Mersenna 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 and of exponent at most 2^n−4.