scholarly journals On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

scholarly journals Skew group rings which are Galois

2000 ◽  
Vol 23 (4) ◽  
pp. 279-283
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Extension ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inner Automorphism ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

LetS*Gbe a skew group ring of a finite groupGover a ringS. It is shown that ifS*Gis anG′-Galois extension of(S*G)G′, whereG′is the inner automorphism group ofS*Ginduced by the elements inG, thenSis aG-Galois extension ofSG. A necessary and sufficient condition is also given for the commutator subring of(S*G)G′inS*Gto be a Galois extension, where(S*G)G′is the subring of the elements fixed under each element inG′.

scholarly journals Integral Group Rings with Nilpotent Unit Groups

1976 ◽  
Vol 28 (5) ◽  
pp. 954-960 ◽  
Author(s):  
César Polcino Milies
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Unit Element ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
The Subject ◽  
A Finite Group

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.The study of the nilpotency of U(RG) has been the subject of several papers.

scholarly journals On central commutator Galois extensions of rings

2000 ◽  
Vol 24 (5) ◽  
pp. 289-294
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Galois Extension ◽  
Structure Theorem ◽  
Projective Group ◽  
Galois Extensions ◽  
Separable Extensions

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

On Coleman Automorphisms of Extensions of Finite Quasinilpotent Groups by Some Groups

2018 ◽  
Vol 25 (02) ◽  
pp. 181-188 ◽  
Author(s):  
Jinke Hai ◽  
Yixin Zhu
Keyword(s):  
Finite Group ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Inner Automorphism ◽  
Coleman Automorphism ◽  
A Finite Group

Let G be an extension of a finite quasinilpotent group by a finite group. It is shown that under some conditions every Coleman automorphism of G is an inner automorphism. The interest in such automorphisms arose from the study of the normalizer problem for integral group rings. Our theorems generalize some well-known results.

scholarly journals Units of group rings of groups of order 16

1993 ◽  
Vol 35 (3) ◽  
pp. 367-379 ◽  
Author(s):  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Abelian Groups ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Group Of Units ◽  
Abelian Case ◽  
A Finite Group

LetGbe a finite group,(ZG) the group of units of the integral group ring ZGand1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively(ZG) for particular groupsG.This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of(ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

Free Subgroups of Units in Group Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 309-312 ◽  
Author(s):  
Jairo Zacarias Gonçalves
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Free Subgroup ◽  
Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Free Subgroups

AbstractIn this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.Some extensions of this results to infinite nilpotent and FC groups are also considered.

Principally quasi-Baer properties of group rings

2012 ◽  
Vol 49 (4) ◽  
pp. 454-465
Author(s):  
Libo Zan ◽  
Jianlong Chen
Keyword(s):  
Finite Group ◽  
Left Ideal ◽  
Group Ring ◽  
Group Rings ◽  
A Finite Group ◽  
Left Annihilator

A ring R is called left p.q.-Baer if the left annihilator of a principal left ideal is generated, as a left ideal, by an idempotent. It is first proved that for a ring R and a group G, if the group ring RG is left p.q.-Baer then so is R; if in condition G is finite then |G|−1 ∈ R. Counterexamples are given to answer the question whether the group ring RG is left p.q.-Baer if R is left p.q.-Baer and G is a finite group with |G|−1 ∈ R. Further, RD∞ is left p.q.-Baer if and only if R is left p.q.-Baer.

scholarly journals Baer and quasi-Baer properties of group rings

2007 ◽  
Vol 83 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Zhong Yi ◽  
Yiqiang Zhou
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Baer Group ◽  
Baer Ring ◽  
Fixed Ring ◽  
A Finite Group ◽  
Left Annihilator

AbstractA ring R is said to be a Baer (respectively, quasi-Baer) ring if the left annihilator of any nonempty subset (respectively, any ideal) of R is generated by an idempotent. It is first proved that for a ring R and a group G, if a group ring RG is (quasi-) Baer then so is R; if in addition G is finite then |G|–1 € R. Counter examples are then given to answer Hirano's question which asks whether the group ring RG is (quasi-) Baer if R is (quasi-) Baer and G is a finite group with |G|–1 € R. Further, efforts have been made towards answering the question of when the group ring RG of a finite group G is (quasi-) Baer, and various (quasi-) Baer group rings are identified. For the case where G is a group acting on R as automorphisms, some sufficient conditions are given for the fixed ring RG to be Baer.

Homological Dimensions of Skew Group Rings

2020 ◽  
Vol 27 (02) ◽  
pp. 319-330
Author(s):  
Yueming Xiang
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Global Dimension ◽  
Group Rings ◽  
Separable Extension ◽  
Homological Dimensions ◽  
A Finite Group ◽  
Coefficient Ring ◽  
Skew Group Ring

Let R be a ring and let H be a subgroup of a finite group G. We consider the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of the skew group ring RσG and its coefficient ring R. Under the assumption that RσG is a separable extension over RσH, it is shown that RσG and RσH share the same homological dimensions. Several known results are then obtained as corollaries. Moreover, we investigate the relationships between the homological dimensions of RσG and the homological dimensions of a commutative ring R, using the trivial RσG-module.

Determining Units in Some Integral Group Rings

1990 ◽  
Vol 33 (2) ◽  
pp. 242-246 ◽  
Author(s):  
E. G. Goodaire ◽  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Dihedral Group ◽  
Group Rings ◽  
Integral Group ◽  
Basic Information ◽  
Integral Group Rings ◽  
Special Cases ◽  
A Finite Group ◽  
Made In

In this brief note, we will show how in principle to find all units in the integral group ring ZG, whenever G is a finite group such that and Z(G) each have exponent 2, 3, 4 or 6. Special cases include the dihedral group of order 8, whose units were previously computed by Polcino Milies [5], and the group discussed by Ritter and Sehgal [6]. Other examples of noncommutative integral group rings whose units have been computed include , but in general very little progress has been made in this direction. For basic information on units in group rings, the reader is referred to Sehgal [7].

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