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 2n 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 2m in R +, then either m is a prime number for which 2m − 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 2m (2m − 1)). In the second case, either n < 7 or the subgroup L is abelian and R * is a nonmetacyclic group of order 2n−1 whose exponent does not exceed 2n−4.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 6, pp. 811–818, June, 2012.
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Raevs’ka, M.Y., Sysak, Y.P. On local near-rings with Miller–Moreno multiplicative group. Ukr Math J 64, 930–937 (2012). https://doi.org/10.1007/s11253-012-0688-z
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DOI: https://doi.org/10.1007/s11253-012-0688-z