Finite local nearrings

Maryna Raievska,Iryna Raievska

doi:10.18523/2617-7080i2018p38-48

Abstract

Nearrings arise naturally in the study of systems of nonlinear mappings, and they have been studied for many decades. Basic definitions and many results concerning nearrings can be, for instance, found in [G. Pilz. Near-rings. The theory and its applications. North Holland, Amsterdam, 1977].Nearrings are generalized rings in the sense that the addition need not be commutative and only one distributive law is assumed. Clearly, every associative ring is a nearring, and each group is an additive group of a nearring, but not necessarily of a nearring with identity. The question what group can be an additive group of a nearring with identity is far from solution.A nearring with identity is called local if the set of all its non-invertible elements is a subgroup of its additive group. A study of local nearrings was initiated by Maxson (1968) who defined a number of their basic properties and proved, in particular, that the additive group of a finite zero-symmetric local nearring is a p-group. The determination of the non-abelian finite p-groups which are the additive groups of local nearrings is an open problem (Feigelstock, 2006).The list of all local nearrings of order at most 31 can be extracted from the package SONATA (https://www.gap-system.org/Packages/sonata.html) of the computer system algebra GAP (https:// www.gap-system.org/).We observe also that there exist 14 non-isomorphic groups of order 16 = 24 from which 9 are the additive groups of local nearrings. Groups of order 32 = 25 with this property are described. In particular, among 51 non-isomorphic groups of this order only 19 are these additive groups.In this paper finite local nearrings are studied. Moreover, local nearrings of order at most 32 are classified.

Highlights

they have been studied for many decades
The question what group can be an additive group of a nearring with identity is
A nearring with identity is called local if the set

Summary

Попереднi результати

Майже-кiльця — це множини з двома бiнарними операцiями, додаванням та множенням, що задовольняють усi аксiоми асоцiативного кiльця, за винятком комутативностi додавання та одного (у нашому випадку, правого) дистрибутивного закону. Нехай адитивна група локального майже-кiльця R iзоморфна групi G(pm, pn, p) для пiдходящих простого p та натуральних чисел m, n, тодi R+ = = a + b + c з елементами a, b та c, що задовольняють спiввiдношення apm = bpn = 0, −b + a + b = a + c та −b + c + b = c, причому a є одиничним елементом в R та кожний елемент x ∈ R однозначно записується у виглядi x = = ax1 + bx2 + cx з коефiцiєнтами 0 ≤ x1 < pm, 0 ≤ x2 < pn та 0 ≤ x3 < p. Нехай n(G) — кiлькiсть усiх неiзоморфних локальних майже-кiлець R порядку не вище 31, якi не є майже-полями, адитивна група R+ яких iзоморфна групi G. Якщо n(R∗) — кiлькiсть цих майже-кiлець, для яких IдГрупи(R∗) є фiксованим, то виконується таке (див. табл. 2)

Структурний IдГрупи опис

Список лiтератури

FINITE LOCAL NEARRINGS