Subgroups induced by certain ideals of free group rings

1983 ◽  
Vol 11 (22) ◽  
pp. 2519-2525 ◽  
Author(s):  
Chander Kanta Gupta
Keyword(s):  
Free Group ◽  
Group Rings

A Problem of R. H. Fox

1981 ◽  
Vol 24 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Group Rings ◽  
Normal Subgroups ◽  
Infinite Groups ◽  
Expository Article

The purpose of this expository article is to familiarize the reader with one of the fundamental problems in the theory of infinite groups. We give an up-to-date account of the so-called Fox problem which concerns the identification of certain normal subgroups of free groups arising out of certain ideals in the free group rings. We assume that the reader is familiar with the elementary concepts of algebra.

scholarly journals Free group rings and derived functors

2018 ◽  
pp. 407-425
Author(s):  
Roman Mikhailov ◽  
Inder Bir Passi
Keyword(s):  
Free Group ◽  
Group Rings ◽  
Derived Functors

Polynomial Ideals in Group Rings

1973 ◽  
Vol 25 (6) ◽  
pp. 1174-1182 ◽  
Author(s):  
M. M. Parmenter ◽  
I. B. S. Passi ◽  
S. K. Sehgal
Keyword(s):  
Commutative Ring ◽  
Free Group ◽  
Group Ring ◽  
Multiplicative Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Polynomial Ideals

Letf(x1, x2, … , xn) be a polynomial in n non-commuting variables x1, x2, … , xn and their inverses with coefficients in the ring Z of integers, i.e. an element of the integral group ring of the free group on X1, x2, … , xn. Let R be a commutative ring with unity, G a multiplicative group and R(G) the group ring of G with coefficients in R.

ESSENTIALITY OF FRACTAL IDEALS

1993 ◽  
Vol 03 (04) ◽  
pp. 425-445
Author(s):  
AMNON ROSENMANN
Keyword(s):  
Free Group ◽  
Group Rings

The question of essentiality of decreasing and increasing fractal ideals is investigated, mostly for fractal ideals built from the agumentation ideal of free group rings. It is shown that this question can be answered in terms of the structure of the trees representing the fractal ideals.

Free Group Rings

1987 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Free Group ◽  
Group Rings

Idempotents in Noetherian Group Rings

1973 ◽  
Vol 25 (2) ◽  
pp. 366-369 ◽  
Author(s):  
Edward Formanek
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Related Result ◽  
Group Rings ◽  
Ordered Group ◽  
Torsion Free Group ◽  
Zero Divisors ◽  
Torsion Free

If G is a torsion–free group and F is a field, is the group ring F[G] a ring without zero divisors? This is true if G is an ordered group or various generalizations thereof - beyond this the question remains untouched. This paper proves a related result.

Zero Divisors and Idempotents in Group Rings

1980 ◽  
Vol 32 (3) ◽  
pp. 596-602 ◽  
Author(s):  
Gerald H. Cliff
Keyword(s):  
Commutative Ring ◽  
Free Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Group Rings ◽  
Torsion Free Group ◽  
Zero Divisors ◽  
Nonzero Characteristic ◽  
Torsion Free ◽  
Major Progress

We consider the following problem: If KG is the group ring of a torsion free group over a field K,show that KG has no divisors of zero. At characteristic zero, major progress was made by Brown [2], who solved the problem for G abelian-by-finite, and then by Farkas and Snider [4], who considered Gpolycyclic-by-finite. Here we present a solution at nonzero characteristic for polycyclic-by-finite groups. We also show that if Khas characteristic p > 0 and G is polycyclic-by-finite with only p-torsion, then KG has no idempotents other than 0 or 1. Finally we show that if R is a commutative ring of nonzero characteristic without nontrivial idempotents and G is polycyclic-by-finite such that no element different from 1 in G has order invertible in R, then RG has no nontrivial idempotents. This is proved at characteristic zero in [3].

Augmentation Quotients of Free Group Rings

2005 ◽  
Vol 12 (04) ◽  
pp. 597-606 ◽  
Author(s):  
Ram Karan ◽  
Deepak Kumar
Keyword(s):  
Free Group ◽  
Group Rings

Let F be a free group and R be a subgroup of F. It is proved that [Formula: see text] are free-abelian. Explicit bases of first two and complete descriptions of all these groups are also given.

scholarly journals On zero divisors with small support in group rings of torsion-free groups

2013 ◽  
Vol 16 (5) ◽  
Author(s):  
Pascal Schweitzer
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Free Groups ◽  
Group Rings ◽  
Torsion Free Group ◽  
Zero Divisors ◽  
Torsion Free ◽  
Small Support

Abstract.Kaplansky's zero divisor conjecture envisions that for a torsion-free group 

scholarly journals GROUP RINGS OF COUNTABLE NON-ABELIAN LOCALLY FREE GROUPS ARE PRIMITIVE

2011 ◽  
Vol 21 (03) ◽  
pp. 409-431 ◽  
Author(s):  
TSUNEKAZU NISHINAKA
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Free Groups ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hnn Extensions ◽  
Necessary And Sufficient ◽  
Countably Infinite

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In addition, by making use of the result, we give a necessary and sufficient condition for group rings of ascending HNN extensions of free groups to be primitive, which extends the main result in [Group rings of proper ascending HNN extensions of countably infinite free groups are primitive, J. Algebra317 (2007) 581–592] to the general cardinality case.

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