Local nearrings with dihedral multiplicative group. (English) Zbl 1048.16027
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 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.
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.
Reviewer: Wen-Fong Ke (Tainan)
MSC:
| 16Y30 | Near-rings |
| 16L30 | Noncommutative local and semilocal rings, perfect rings |
| 16U60 | Units, groups of units (associative rings and algebras) |
| 20D40 | Products of subgroups of abstract finite groups |
Citations:
Zbl 0159.03902References:
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