A NOTE ON GROUP RINGS WITH TRIVIAL UNITS

2021 ◽  
pp. 1-5
Author(s):  
A. Y. M. CHIN
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Torsion Group ◽  
Locally Finite Group ◽  
Group Rings ◽  
Prime Characteristic ◽  
Locally Finite ◽  
Characteristic Two

Abstract Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

Cleanness of the Group Ring of an Abelian p-Group over a Commutative Ring

2012 ◽  
Vol 19 (03) ◽  
pp. 539-544
Author(s):  
Xiulan Wang ◽  
Hong You
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Locally Finite Group ◽  
Locally Finite

A ring R is called clean if every element is the sum of an idempotent and a unit, while R is called uniquely clean if this representation is unique. In this article, we prove that if R is a commutative ring and G is an abelian p-group with p in J(R), then RG is clean if and only if R is clean. Moreover, when G is a locally finite group, some conditions for RG to be uniquely clean are given.

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.

Trivial Units for Group Rings with G-adapted Coefficient Rings

2005 ◽  
Vol 48 (1) ◽  
pp. 80-89 ◽  
Author(s):  
Allen Herman ◽  
Yuanlin Li ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Finite Characteristic

AbstractFor each finite group G for which the integral group ring ℤG has only trivial units, we give ring-theoretic conditions for a commutative ring R under which the group ring RG has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if R is a ring of finite characteristic and RG has only trivial units, then G has order at most 3.

scholarly journals STRONGLY CLEAN POWER SERIES RINGS

2007 ◽  
Vol 50 (1) ◽  
pp. 73-85 ◽  
Author(s):  
Jianlong Chen ◽  
Yiqiang Zhou
Keyword(s):  
Finite Group ◽  
Power Series ◽  
Group Ring ◽  
Regular Ring ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Regular Elements ◽  
Strongly Regular ◽  
Power Series Rings

AbstractAn element $a$ in a ring $R$ with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And $a\in R$ is called strongly $\pi$-regular if both chains $aR\supseteq a^2R\supseteq\cdots$ and $Ra\supseteq Ra^2\supseteq\cdots$ terminate. A ring $R$ is called strongly clean (respectively, strongly $\pi$-regular) if every element of $R$ is strongly clean (respectively, strongly $\pi$-regular). Strongly $\pi$-regular elements of a ring are all strongly clean. Let $\sigma$ be an endomorphism of $R$. It is proved that for $\varSigma r_ix^i\in R[[x,\sigma]]$, if $r_0$ or $1-r_0$ is strongly $\pi$-regular in $R$, then $\varSigma r_ix^i$ is strongly clean in $R[[x,\sigma]]$. In particular, if $R$ is strongly $\pi$-regular, then $R[[x,\sigma]]$ is strongly clean. It is also proved that if $R$ is a strongly $\pi$-regular ring, then $R[x,\sigma]/(x^n)$ is strongly clean for all $n\ge1$ and that the group ring of a locally finite group over a strongly regular or commutative strongly $\pi$-regular ring is strongly clean.

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.

Semigroup identities on units of integral group rings

1997 ◽  
Vol 39 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Jairo Z. Gonçalves
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Group Ring ◽  
Torsion Group ◽  
Group Rings ◽  
Integral Group ◽  
Semigroup Identity ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Semigroup Identities

AbstractLet U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.

Groups with every subgroup ascendant-by-finite

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Sergio Camp-Mora
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Nilpotent

AbstractA subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

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.

scholarly journals Group rings which are v-HC orders and Krull orders

1991 ◽  
Vol 34 (2) ◽  
pp. 217-228 ◽  
Author(s):  
K. A. Brown ◽  
H. Marubayashi ◽  
P. F. Smith
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Prime Ideals ◽  
Group Rings

Let R be a ring and G a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and G, when the group ring R[G] is a prime Krull order and when it is a price v-HC order. The key ingredient in obtaining both characterizations is the first author's earlier study of height one prime ideals in the ring R[G[.

Centralizers of subgroups in simple locally finite groups

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Kıvanç Ersoy ◽  
Mahmut Kuzucuoğlu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Subgroup ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Non Linear

AbstractHartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely letMoreover we prove that if

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