On the structure of the Group-ring of an elementary abelian group and a Theorem of Brauer and Nesbitt

1981 ◽  
Vol 67 (1) ◽  
pp. 260-267
Author(s):  
Prabir Bhattacharya
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Elementary Abelian Group

Functors on Finite Vector Spaces and Units in Abelian Group Rings

1986 ◽  
Vol 29 (1) ◽  
pp. 79-83 ◽  
Author(s):  
Klaus Hoechsmann
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Elementary Abelian Group ◽  
Vector Spaces ◽  
Group Rings ◽  
Cyclic Subgroups ◽  
Group Of Units ◽  
Finite Vector ◽  
Finite Vector Spaces

AbstractIf A is an elementary abelian group, let (A) denote the group of units, modulo torsion, of the group ring Z[A]. We study (A) by means of the compositewhere C and B run over all cyclic subgroups and factor-groups, respectively. This map, γ, is known to be injective; its index, too, is known. In this paper, we determine the rank of γ tensored (over Z);with various fields. Our main result depends only on the functoriality of

Cyclotomic units and the unit group of an elementary abelian group ring

1985 ◽  
Vol 45 (1) ◽  
pp. 5-7 ◽  
Author(s):  
K. Hoechsmann ◽  
S. K. Sehgal ◽  
A. Weiss
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Unit Group ◽  
Elementary Abelian Group ◽  
Abelian Group Ring ◽  
Cyclotomic Units

scholarly journals The number of idempotents in abelian group rings

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 719-723
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Group Rings ◽  
Arbitrary Characteristic

Suppose that R is a commutative unitary ring of arbitrary characteristic and G is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set id(RG), consisting of all idempotent elements in the group ring RG. It is explicitly calculated only in terms associated with R, G and their divisions. This result strengthens previous estimates obtained in the literature recently.

Outer Partial Actions and Partial Skew Group Rings

2015 ◽  
Vol 67 (5) ◽  
pp. 1144-1160 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Group Ring ◽  
Group Rings ◽  
Partial Action ◽  
Unital Ring ◽  
Different Types ◽  
Outer Action ◽  
Skew Group Rings ◽  
Skew Group Ring

AbstractWe extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

scholarly journals Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

The Group of Units of the Integral Group Ring ZS3

1972 ◽  
Vol 15 (4) ◽  
pp. 529-534 ◽  
Author(s):  
I. Hughes ◽  
K. R. Pearson
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Torsion Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
List Type ◽  
Type Number ◽  
Quaternion Group ◽  
Group Of Units

We denote by ZG the integral group ring of the finite group G. We call ±g, for g in G, a trivial unit of ZG. For G abelian, Higman [4] (see also [3, p. 262 ff]) showed that every unit of finite order in ZG is trivial. For arbitrary finite G (indeed, for a torsion group G, not necessarily finite), Higman [4] showed that every unit in ZG is trivial if and only if G is(i) abelian and the order of each element divides 4, or(ii) abelian and the order of each element divides 6, or(iii) the direct product of the quaternion group of order 8 and an abelian group of exponent 2.

scholarly journals The Noether number of p-groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950066 ◽  
Author(s):  
Kálmán Cziszter
Keyword(s):  
Abelian Group ◽  
Heisenberg Group ◽  
Cyclic Subgroup ◽  
Elementary Abelian Group ◽  
Polynomial Invariant

A group of order [Formula: see text] ([Formula: see text] prime) has an indecomposable polynomial invariant of degree at least [Formula: see text] if and only if the group has a cyclic subgroup of index at most [Formula: see text] or it is isomorphic to the elementary abelian group of order 8 or the Heisenberg group of order 27.

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