Local nearrings with dihedral multiplicative group. (Q1427421)

A nearring \(R\) is said to be local if \(R\) has a unit element and the set \(L_R\) of non-invertible elements of \(R\) forms an additive subgroup of \(R\). It was shown by \textit{C. J. Maxson} [in Math. Z. 106, 197-205 (1968; Zbl 0159.03902] that if \(R\) is a finite local nearring, then \((R,+)\) is a \(p\)-group for some prime \(p\). In this article, the authors show that if \(R\) is a local nearring with the group of units \(R^*\) being dihedral, then \(R\) is finite. \((R,+)\) is either a 3-group of order at most 9, or a 2-group of order at most 32. \(L_R\) is either an Abelian group or a group of order 16 with derived subgroup of order 2.