A Problem of Herstein on Group Rings

1974 ◽  
Vol 17 (2) ◽  
pp. 201-202 ◽  
Author(s):  
Edward Formanek
Keyword(s):  
Group Ring ◽  
Finite Subset ◽  
Zero Divisor ◽  
Formal Proof ◽  
Left Zero ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position ◽  
Finite Sum

Let F be a field of characteristic 0 and G a group such that each element of the group ring F[G] is either (right) invertible or a (left) zero divisor. Then G is locally finite.This answers a question of Herstein [1, p. 36] [2, p. 450] in the characteristic 0 case. The proof can be informally summarized as follows: Let gl,…,gn be a finite subset of G, and let1—x is not a zero divisor so it is invertible and its inverse is 1+x+x3+⋯. The fact that this series converges to an element of F[G] (a finite sum) forces the subgroup generated by g1,…,gn to be finite, proving the theorem. The formal proof is via epsilontics and takes place inside of F[G].

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.

On the structure of some Noetherian modules over group rings

2014 ◽  
Vol 24 (02) ◽  
pp. 233-249
Author(s):  
Leonid A. Kurdachenko ◽  
Javier Otal ◽  
Igor Ya. Subbotin
Keyword(s):  
Finite Field ◽  
Group Ring ◽  
Soluble Group ◽  
Dedekind Domain ◽  
Group Rings ◽  
Property A ◽  
Locally Finite ◽  
Locally Finite Field ◽  
Noetherian Modules

In this paper, we study the structure of some Noetherian modules over group rings and deduce some statements regarding the structure of the groups involved. More precisely, we consider a module A over a group ring RG with the following property: A is a Noetherian RH-module for every subgroup H, which is not contained in the centralizer CG(A). If G is some generalized soluble group and R is a locally finite field or some Dedekind domain, we describe the structure of G/CG(A).

A Note on Derivations of Group Rings

1995 ◽  
Vol 38 (4) ◽  
pp. 434-437 ◽  
Author(s):  
Miguel Ferrero ◽  
Antonio Giambruno ◽  
César Polcino Milies
Keyword(s):  
Group Ring ◽  
Finite Index ◽  
Semiprime Ring ◽  
Group Rings ◽  
Locally Finite

AbstractLetRGdenote the group ring of a groupGover a semiprime ringR. We prove that, if the center ofGis of finite index and some natural restrictions hold, then everyR-derivation ofRGis inner. We also give an example of a groupGwhich is both locally finite and nilpotent and such that, for every fieldF, there exists anF-derivation ofFGwhich is not inner.

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.

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].

Zero divisors and idempotents in group rings

1977 ◽  
Vol 81 (3) ◽  
pp. 365-368 ◽  
Author(s):  
P. A. Linnell
Keyword(s):  
Group Algebra ◽  
Zero Divisor ◽  
Affirmative Answer ◽  
Supersoluble Group ◽  
Group Rings ◽  
Prime Characteristic ◽  
Locally Finite ◽  
Zero Divisors ◽  
Torsion Free

1. Introduction. Let kG denote the group algebra of a group G over a field k. In this paper we are first concerned with the zero divisor conjecture: that if G is torsion free, then kG is a domain. Recently, K. A. Brown made a remarkable breakthrough when he settled the conjecture for char k = 0 and G abelian by finite (2). In a beautiful paper written shortly afterwards, D. Farkas and R. Snider extended this result to give an affirmative answer to the conjecture for char k = 0 and G polycyclic by finite. Their methods, however, were less successful when k had prime characteristic. We use the techniques of (4) and (7) to prove the following:Theorem A. If G is a torsion free abelian by locally finite by supersoluble group and k is any field, then kG is a domain.

The Periodic Radical of Group Rings and Incidence Algebras

1998 ◽  
Vol 41 (4) ◽  
pp. 481-487 ◽  
Author(s):  
M. M. Parmenter ◽  
E. Spiegel ◽  
P. N. Stewart
Keyword(s):  
Group Ring ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Locally Finite ◽  
Partially Ordered ◽  
Incidence Algebras ◽  
Finite Partially Ordered Set ◽  
Necessary And Sufficient

AbstractLet R be a ring with 1 and P(R) the periodic radical of R. We obtain necessary and sufficient conditions for P(RG) = 0 when RG is the group ring of an FC group G and R is commutative. We also obtain a complete description of when I(X, R) is the incidence algebra of a locally finite partially ordered set X and R is commutative.

Central Idempotents in Group Rings

1970 ◽  
Vol 13 (4) ◽  
pp. 527-528 ◽  
Author(s):  
R. G. Burns
Keyword(s):  
Support Group ◽  
Group Ring ◽  
Group Rings ◽  
Finite Support ◽  
Image Position

Let R be a ring and G a group. The group ring RG consists of all functions f: G → R with finite support. Addition is pointwise and multiplication is defined for f, h ∊ RG and g ∊ G, byThe support group of f is defined to be the subgroup of G generated by the support of f. The element f is idempotent if ff = fWe prove the following result.

Zero-divisor graph of semisimple group-rings

2020 ◽  
pp. 2250028
Author(s):  
Krishnan Paramasivam ◽  
K. Muhammed Sabeel
Keyword(s):  
Commutative Ring ◽  
Structural Properties ◽  
Cyclic Group ◽  
Group Ring ◽  
Zero Divisor ◽  
Semisimple Group ◽  
Group Rings ◽  
Zero Divisor Graph ◽  
Finite Set

Let [Formula: see text], [Formula: see text], [Formula: see text] denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring [Formula: see text], respectively. In this paper, we prove that [Formula: see text] for a semisimple commutative ring [Formula: see text] and represent [Formula: see text] as a generalized join of a finite set of graphs. Further, we study the zero-divisor graph of a semisimple group-ring [Formula: see text] and proved several structural properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is a field with [Formula: see text] elements and [Formula: see text] is a cyclic group with [Formula: see text] elements.

Nil-clean group rings

2016 ◽  
Vol 16 (07) ◽  
pp. 1750135 ◽  
Author(s):  
Serap Sahinkaya ◽  
Gaohua Tang ◽  
Yiqiang Zhou
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Ring ◽  
Nilpotent Element ◽  
General Situation ◽  
Group Rings ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Clean Ring ◽  
Nil Clean Ring

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a [Formula: see text]-group.

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