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].

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 Some torsion-free group rings with nilpotent prime radicals

1974 ◽  
Vol 18 (3) ◽  
pp. 372-375 ◽  
Author(s):  
Keng-Teh Tan
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Group Rings ◽  
Torsion Free Group ◽  
Torsion Free

LetRbe a ring with identity. We will useJ(R) andP(R) to denote the Jacobson and prime radicals ofR, respectively. IfGis a group, the group ring ofGoverRwill be denoted byRG.

A Note on Group Rings of Certain Torsion-Free Groups

1972 ◽  
Vol 15 (3) ◽  
pp. 441-445 ◽  
Author(s):  
R. G. Burns ◽  
V. W. D. Hale
Keyword(s):  
Group Ring ◽  
Analogous Result ◽  
Free Groups ◽  
Group Rings ◽  
Finitely Generated ◽  
Zero Divisors ◽  
Epimorphic Image ◽  
Torsion Free

AbstractAs a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.

scholarly journals A just-nonsolvable torsion-free group defined on the binary tree

1999 ◽  
Vol 211 (1) ◽  
pp. 99-114 ◽  
Author(s):  
A.M. Brunner ◽  
Said Sidki ◽  
Ana Cristina Vieira
Keyword(s):  
Free Group ◽  
Binary Tree ◽  
Torsion Free Group ◽  
Torsion Free

scholarly journals On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

scholarly journals Abelian groups with small cotorsion images

1991 ◽  
Vol 50 (2) ◽  
pp. 243-247 ◽  
Author(s):  
Rüdiger Göbel
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Abelian Groups ◽  
Torsion Free Group ◽  
Compact Abelian Groups ◽  
Torsion Free

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

scholarly journals ON THE ASPHERICITY OF LENGTH-6 RELATIVE PRESENTATIONS WITH TORSION-FREE COEFFICIENTS

2008 ◽  
Vol 51 (1) ◽  
pp. 201-214
Author(s):  
Seong Kun Kim
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Point Of View ◽  
Torsion Free Group ◽  
Torsion Free

AbstractAn interesting result of Ivanov implies that a non-aspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture. In this point of view, we prove the asphericity of the length-6 relative presentation $\langle H,x: xh_1xh_2xh_3xh_4xh_5xh_6\rangle$, provided that each coefficient is torsion free.

Periodic cohomology and subgroups with bounded Bredon cohomological dimension

2008 ◽  
Vol 144 (2) ◽  
pp. 329-336 ◽  
Author(s):  
JANG HYUN JO ◽  
BRITA E. A. NUCINKIS
Keyword(s):  
Free Group ◽  
Amenable Group ◽  
Cohomological Dimension ◽  
Dimensional Model ◽  
Torsion Free Group ◽  
Bredon Cohomology ◽  
Finite Dimensional ◽  
Torsion Free

AbstractMislin and Talelli showed that a torsion-free group in$\HF$with periodic cohomology after some steps has finite cohomological dimension. In this note we look at similar questions for groups with torsion by considering Bredon cohomology. In particular we show that every elementary amenable group acting freely and properly on some$\R^n$×Smadmits a finite dimensional model for$\E$G.

scholarly journals Torsion-free group without unique product property

1987 ◽  
Vol 108 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Eliyahu Rips ◽  
Yoav Segev
Keyword(s):  
Free Group ◽  
Torsion Free Group ◽  
Product Property ◽  
Unique Product ◽  
Torsion Free
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