scholarly journals Computing generators of the unit group of an integral abelian group ring

2013 ◽  
Vol 373 ◽  
pp. 441-452 ◽  
Author(s):  
Paolo Faccin ◽  
Willem A. de Graaf ◽  
Wilhelm Plesken
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Unit Group ◽  
Abelian Group Ring

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 Full Identification of Idempotens in Binary Abelian Group Rings

2017 ◽  
Vol 23 (2) ◽  
pp. 67-75
Author(s):  
Kai Lin Ong ◽  
Miin Huey Ang
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Group Ring ◽  
Minimal Distance ◽  
Group Rings ◽  
Cyclic Groups ◽  
Algebraic Properties ◽  
Abelian Group Ring ◽  
Binary Group

Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent.

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.

TRIVIAL TORSION UNITS IN G-ADAPTED GROUP RINGS

2006 ◽  
Vol 05 (06) ◽  
pp. 781-791
Author(s):  
ALLEN HERMAN ◽  
YUANLIN LI
Keyword(s):  
Group Ring ◽  
Torsion Group ◽  
Unit Group ◽  
Group Rings ◽  
Quaternion Group ◽  
Torsion Units ◽  
Equivalent Conditions

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ8 has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

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.

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

scholarly journals INTEGRAL GROUP RING OF THE MATHIEU SIMPLE GROUP M24

2012 ◽  
Vol 11 (01) ◽  
pp. 1250016 ◽  
Author(s):  
VICTOR BOVDI ◽  
ALEXANDER KONOVALOV
Keyword(s):  
Simple Group ◽  
Group Ring ◽  
Unit Group ◽  
Positive Answer ◽  
Integral Group Ring ◽  
Integral Group ◽  
Sporadic Group ◽  
Prime Graphs ◽  
Zassenhaus Conjecture

We study the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M24. As a consequence, for this group we give a positive answer to the question by Kimmerle about prime graphs.

INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS

2011 ◽  
Vol 10 (04) ◽  
pp. 711-725 ◽  
Author(s):  
J. Z. GONÇALVES ◽  
D. S. PASSMAN
Keyword(s):  
Group Ring ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Nonabelian Group ◽  
Ring Of Integers ◽  
Integral Group Rings ◽  
Noncyclic Subgroup

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (uk, m(x), uk, m(x*)) or (uk, m(x), uk, m(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

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