On the Uniqueness of the Coefficient Ring in a Group Ring

1983 ◽  
Vol 35 (4) ◽  
pp. 654-673 ◽  
Author(s):  
Isabelle Adjaero ◽  
Eugene Spiegel
Keyword(s):  
Commutative Ring ◽  
Cyclic Group ◽  
Group Ring ◽  
Commutative Rings ◽  
Infinite Cyclic Group ◽  
Ring Of Polynomials ◽  
Image Position ◽  
Coefficient Ring

Let R1 and R2 be commutative rings with identities, G a group and R1G and R2G the group ring of G over R1 and R2 respectively. The problem that motivates this work is to determine what relations exist between R1 and R2 if R1G and R2G are isomorphic. For example, is the coefficient ring R1 an invariant of R1G? This is not true in general as the following example shows. Let H be a group andIf R1 is a commutative ring with identity and R2 = R1H, thenbut R1 needn't be isomorphic to R2.Several authors have investigated the problem when G = <x>, the infinite cyclic group, partly because of its closeness to R[x], the ring of polynomials over R.

Uniqueness of the Coefficient Ring in Some Group Rings

1973 ◽  
Vol 16 (4) ◽  
pp. 551-555 ◽  
Author(s):  
M. Parmenter ◽  
S. Sehgal
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Jacobson Radical ◽  
Group Rings ◽  
Infinite Cyclic Group ◽  
Coefficient Ring

Let 〈x〉 be an infinite cyclic group and Ri〈x〉 its group ring over a ring (with identity) Ri, for i = l and 2. Let J(Ri) be the Jacobson radical of Ri. In this note we study the question of whether or not R1〈x〉≃R2〈x〉 implies R1≃R2. We prove that this is so if Zi the centre of Ri is semi-perfect and J(Zi〈x〉) = J(Zi〈)x〉 for i = l and 2. In particular, when Zi is perfect the second condition is satisfied and the isomorphism of group rings Ri〈x〉 implies the isomorphism of Ri.

Lehmer’s problem for arbitrary groups

2020 ◽  
pp. 1-32
Author(s):  
W. Lück
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Bounded Operator ◽  
Integral Group Ring ◽  
Integral Group ◽  
Infinite Cyclic Group ◽  
Matrix Formula ◽  
Lehmer’S Problem

We consider the problem whether for a group [Formula: see text] there exists a constant [Formula: see text] such that for any [Formula: see text]-matrix [Formula: see text] over the integral group ring [Formula: see text] the Fuglede–Kadison determinant of the [Formula: see text]-equivariant bounded operator [Formula: see text] given by right multiplication with [Formula: see text] is either one or greater or equal to [Formula: see text]. If [Formula: see text] is the infinite cyclic group and we consider only [Formula: see text], this is precisely Lehmer’s problem.

Isomorphic Group Rings with Non-Isomorphic Coefficient Rings*

1980 ◽  
Vol 23 (2) ◽  
pp. 245-246 ◽  
Author(s):  
L. Grünenfelder ◽  
M. M. Parmenter
Keyword(s):  
Cyclic Group ◽  
Research Problem ◽  
Commutative Rings ◽  
Group Rings ◽  
Infinite Cyclic Group ◽  
Isomorphic Group ◽  
Unital Rings

The following question has been floating around for some time now and is also stated as Research Problem 26 in [4]:Let R, S be unital rings and let 〈x〉 be an infinite cyclic group. Does R〈x〉≃S〈x〉 imply R≃S?In this note, we present a collection of examples which answer the question in the negative. However, all of these examples consist of non-commutative rings, and the problem is still open in the case where R and S are assumed to be commutative.

On the Galois Theory of Commutative Rings II: Automorphisms induced in Residue Rings

1987 ◽  
Vol 39 (5) ◽  
pp. 1025-1037
Author(s):  
Carl Faith
Keyword(s):  
Commutative Ring ◽  
Galois Theory ◽  
Commutative Rings ◽  
Group Of Automorphisms ◽  
Residue Ring ◽  
Image Position

Let G be a group of automorphisms of a commutative ring K, and let KG denote the Galois subring consisting of all elements left fixed by every g in G. An ideal M is G-stable, or G-invariant, provided that g(x) lies in M for every x in M, that is, g(M) ⊆ M, for every g in G. Then, every g in G induces an automorphism in the residue ring , and if is the group consisting of all , trivially1When the inclusion (1) is strict, then G is said to be cleft at M, or by M, and otherwise G is uncleft at (by) M. When G is cleft at all ideals except 0, then G is cleft, and uncleft otherwise.

On Group Rings

1970 ◽  
Vol 22 (2) ◽  
pp. 249-254 ◽  
Author(s):  
D. B. Coleman
Keyword(s):  
Nilpotent Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Characteristic Zero ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Closed Field ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

Partition Rings of Cyclic Groups of Odd Prime Power Order1

1961 ◽  
Vol 13 ◽  
pp. 373-391
Author(s):  
K. I. Appel
Keyword(s):  
Commutative Ring ◽  
Group Ring ◽  
Prime Power ◽  
Group Rings ◽  
Ring Of Integers ◽  
Cyclic Groups ◽  
Image Position

A ring R over a commutative ring K, that has a basis of elements g1, g2, … , gn forming a group G under multiplication, is called a group ring of G over K. Since all group rings of a given G over a given K are isomorphic, we may speak of the group ring KG of G over X.Let π be any partition of G into non-empty sets GA, GB, … . Any subring P of KG that has a basis of elementsis a partition ring of G over K.If P is a partition ring of G over Z, the ring of integers, then the basis A, B, … for P clearly serves as a basis for a partition ring P’ = Q ⊗ P of G over Q, the field of rationals.

Groups acting freely on R-trees

1991 ◽  
Vol 11 (4) ◽  
pp. 737-756 ◽  
Author(s):  
John W. Morgan ◽  
Richard K. Skora
Keyword(s):  
Euler Characteristic ◽  
Cyclic Group ◽  
Abelian Subgroup ◽  
Closed Surface ◽  
Infinite Cyclic Group ◽  
Precise Result ◽  
Finitely Presented Group ◽  
Hnn Extension ◽  
Image Position ◽  
Finitely Presented

AbstractIn this paper we study the question of which groups act freely on R-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case.Let the finitely presented group G act freely on an R-tree. If A is a non-cyclic, abelian subgroup of G, then A is contained in an abelian subgroup A′ which is a free factor of G.The second part of the paper concerns groups whch split as an HNN-extension along an infinite cyclic group. Here is one formulation of our main result in that case.Let the finitely presented group G act freely on an R-tree. If G has an HNN-decompositionwhere (s) is infinite cyclic, then there is a subgroup H′ ⊂ H such that either(a); or(b),where S is a closed surface of non-positive Euler characteristic.A slightly different, more precise result is also given.

Group Rings over Z(p) with FC Unit Groups

1980 ◽  
Vol 32 (5) ◽  
pp. 1266-1269 ◽  
Author(s):  
H. Merklen ◽  
C. Polcino Milies
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Group Ring ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Conjugacy Classes ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.In [6], S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in [5] and G. Cliffs and S. K. Sehgal [1] completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in [4].

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.

scholarly journals Whitehead groups of semidirect products of free groups

1978 ◽  
Vol 19 (2) ◽  
pp. 155-158 ◽  
Author(s):  
Koo-Guan Choo
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Semidirect Product ◽  
Class Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Whitehead Group ◽  
Infinite Cyclic Group ◽  
Semidirect Products ◽  
Projective Class

Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring ℤ(G) of G by . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

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