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 

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 SOLVABILITY OF CERTAIN EQUATIONS OF ARBITRARY LENGTH OVER TORSION-FREE GROUPS

2020 ◽  
pp. 1-9
Author(s):  
MUHAMMAD FAZEEL ANWAR ◽  
MAIRAJ BIBI ◽  
MUHAMMAD SAEED AKRAM
Keyword(s):  
Free Group ◽  
Zero Divisor ◽  
Free Groups ◽  
Arbitrary Length ◽  
Torsion Free Group ◽  
Torsion Free ◽  
Single Relation

Abstract Let G be a nontrivial torsion-free group and $s\left( t \right) = {g_1}{t^{{\varepsilon _1}}}{g_2}{t^{{\varepsilon _2}}} \ldots {g_n}{t^{{\varepsilon _n}}} = 1\left( {{g_i} \in G,{\varepsilon_i} = \pm 1} \right)$ be an equation over G containing no blocks of the form ${t^{- 1}}{g_i}{t^{ - 1}},{g_i} \in G$ . In this paper, we show that $s\left( t \right) = 1$ has a solution over G provided a single relation on coefficients of s(t) holds. We also generalize our results to equations containing higher powers of t. The later equations are also related to Kaplansky zero-divisor conjecture.

ON THE ASPHERICITY OF CERTAIN RELATIVE PRESENTATIONS OVER TORSION-FREE GROUPS

2008 ◽  
Vol 18 (06) ◽  
pp. 979-987 ◽  
Author(s):  
SEONG KUN KIM
Keyword(s):  
Free Group ◽  
Infinite Order ◽  
Zero Divisor ◽  
Free Groups ◽  
Group Presentation ◽  
Torsion Free Group ◽  
Proper Power ◽  
Torsion Free

It is noteworthy to find whether a relative presentation with torsion-free coefficients is aspherical or not, since a nonaspherical relative presentation that defines a torsion-free group would provide a potential counterexample to the Kaplansky zero-divisor conjecture, see [7]. In this paper we investigate asphericity of the relative group presentation [Formula: see text] provided that m1, m2, …, mk are nonzero integers, k ≤ 3, each coefficient has infinite order and the relator is not a proper power.

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 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.

scholarly journals On singular equations over torsion-free groups

2021 ◽  
pp. 1-30
Author(s):  
Martin Edjvet ◽  
James Howie
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Singular Equation ◽  
Torsion Free Group ◽  
Singular Equations ◽  
Torsion Free

We prove a Freiheitssatz for one-relator products of torsion-free groups, where the relator has syllable length at most [Formula: see text]. This result has applications to equations over torsion-free groups: in particular a singular equation of syllable length at most [Formula: see text] over a torsion-free group has a solution in some overgroup.

ON SMALL LENGTH EQUATIONS OVER TORSION-FREE GROUPS

1994 ◽  
Vol 04 (04) ◽  
pp. 575-589 ◽  
Author(s):  
MATVEĬ I. PRISHCHEPOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Small Length ◽  
Torsion Free Group ◽  
The Absolute ◽  
Torsion Free

Let G be a group, <t> the free group generated by t and let r(t)∈G*<t>. The equation r(t)=1 is said to have a solution over G if it has a solution in some group that contains G. There is a conjecture (attributed to F. Levin) that if G is a torsion-free group, then any equation has a solution over G. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of t in r(t) is not greater than six.

scholarly journals Solving equations of length seven over torsion-free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 147-164
Author(s):  
Mairaj Bibi ◽  
Martin Edjvet
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Solving Equations ◽  
Torsion Free

AbstractPrishchepov [16] proved that all equations of length at most six over torsion-free groups are solvable. A different proof was given by Ivanov and Klyachko in [12]. This supports the conjecture stated by Levin [15] that any equation over a torsion-free group is solvable. Here it is shown that all equations of length seven over torsion-free groups are solvable.

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.

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