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].

scholarly journals COMMUTATIVITY OF SKEW SYMMETRIC ELEMENTS IN GROUP RINGS

2007 ◽  
Vol 50 (1) ◽  
pp. 37-47 ◽  
Author(s):  
Osnel Broche Cristo ◽  
César Polcino Milies
Keyword(s):  
Commutative Ring ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

AbstractLet $R$ be a commutative ring with unity and let $G$ be a group. The group ring $RG$ has a natural involution that maps each element $g\in G$ to its inverse. We denote by $RG^-$ the set of skew symmetric elements under this involution. We study necessary and sufficient conditions for $RG^-$ to be commutative.

Skew-symmetric Elements in Nonlinear Involutions in Group Rings

2015 ◽  
Vol 22 (02) ◽  
pp. 321-332
Author(s):  
A. P. Raposo
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient ◽  
Nonlinear Extensions

Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessary and sufficient conditions are given on the group G, its involution, the ring R and the extension for the set of skew-symmetric elements to be commutative and for it to be anticommutative.

Symmetric Elements of Nonlinear Involutions in Group Rings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1041-1050 ◽  
Author(s):  
R. García-Delgado ◽  
A. P. Raposo
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Linear Extensions ◽  
Necessary And Sufficient

Given an involution φ : G → G in a group G and a ring R, we study the extensions, not necessarily linear, to an involution ψ : RG → RG in the group ring RG. We investigate the symmetric elements, those α ∈ RG for which ψ(α) = α, and give necessary and sufficient conditions for the set of symmetric elements, (RG)ψ, to be a subring of RG. This work is a generalization of [6] and references therein where only linear extensions of the group involution are considered.

COMMUTATIVE TALL RINGS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350129
Author(s):  
TOMÁŠ PENK ◽  
JAN ŽEMLIČKA
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Theoretical Criterion ◽  
Necessary And Sufficient

A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.

scholarly journals Unique factorisation in P.I. group-rings

1995 ◽  
Vol 59 (2) ◽  
pp. 232-243 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Necessary And Sufficient

AbstractWe shall give necessary and sufficient conditions on the ring R and the group G for the group-ring RG to be a prime P. I. ring with the unique factorisation property as defined in [5].

scholarly journals SIMPLE REGULAR SKEW GROUP RINGS

2005 ◽  
Vol 04 (02) ◽  
pp. 127-137 ◽  
Author(s):  
KATHI CROW
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Skew Group Rings ◽  
Necessary And Sufficient ◽  
Skew Group Ring

Given a group G acting on a ring R via α:G → Aut (R), one can construct the skew group ring R*αG. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain a more efficient proof of the simplicity of a certain ring constructed by J. Trlifaj.

BRAUER PAIRS OF VZ-GROUPS

2008 ◽  
Vol 07 (05) ◽  
pp. 663-670 ◽  
Author(s):  
ADRIANA NENCIU
Keyword(s):  
Finite Groups ◽  
Sufficient Conditions ◽  
Conjugacy Classes ◽  
Necessary And Sufficient Conditions ◽  
Irreducible Characters ◽  
Necessary And Sufficient ◽  
Brauer Pairs

Two non-isomorphic finite groups form a Brauer pair if there exist a bijection for the conjugacy classes and a bijection for the irreducible characters that preserve all the character values and the power map. A group is called a VZ-group if all its nonlinear irreducible characters vanish off the center. In this paper we give necessary and sufficient conditions for two non-isomorphic VZ-groups to form a Brauer pair.

scholarly journals Normal and subnormal subgroups in the group of units of group rings

1985 ◽  
Vol 31 (3) ◽  
pp. 355-363 ◽  
Author(s):  
Jairo Zacarias Goncalves
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Finite Subgroup ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Infinite Field ◽  
Group Of Units ◽  
Necessary And Sufficient ◽  
Subnormal Subgroups

Let KG be the group ring of the group G over the infinite field K, and let U(KG) be its group of units. If G is torsion, we obtain necessary and sufficient conditions for a finite subgroup H of G to be either normal or subnormal in U(KG). Actually, if H is subnormal in U(KG), we can handle not only the case H finite, but the precise assumptions depend on the characteristic of K.

scholarly journals Partial actions and cyclic kummer’s theory

2020 ◽  
pp. 2250032
Author(s):  
Víctor Marín ◽  
Andrés Cañas ◽  
Héctor Pinedo
Keyword(s):  
Cyclic Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Galois Extensions ◽  
Radical Extension ◽  
Necessary And Sufficient

We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial [Formula: see text]-Kummerian extension is equivalent to either a radical or an [Formula: see text]-radical extension, for some subgroup [Formula: see text] of the cyclic group [Formula: see text].

Almost Multiplication Rings

1965 ◽  
Vol 17 ◽  
pp. 267-277 ◽  
Author(s):  
H. S. Butts ◽  
R. C. Phillips
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Sufficient Conditions ◽  
Unit Element ◽  
Ideal Theory ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Dedekind Domains ◽  
Necessary And Sufficient ◽  
The Ideal

It is well known that an idealAin a. Dedekind domain has a prime radical if and only ifAis a power of a prime ideal. The purpose of this paper is to determine necessary and sufficient conditions in order that a commutative ring with unit element have this property and to study the ideal theory in such rings. Domains with unit element having the above property possess many of the characteristics of Dedekind domains (however, they need not be Noetherian) and will be referred to in this paper as "almost Dedekind domains"—these domains are considered in Section 1.

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