The Cohn localization of the free group ring

1992 ◽  
Vol 111 (3) ◽  
pp. 433-443 ◽  
Author(s):  
M. Farber ◽  
P. Vogel
Keyword(s):  
Free Group ◽  
Group Ring

In [1] P. Cohn suggested the construction of a localization of a ring with respect to a class of square matrices. Let us briefly recall the definitions.

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.

BASS CYCLIC UNITS AS FACTORS IN A FREE GROUP IN INTEGRAL GROUP RING UNITS

2011 ◽  
Vol 21 (04) ◽  
pp. 531-545 ◽  
Author(s):  
JAIRO Z. GONÇALVES ◽  
ÁNGEL DEL RÍO
Keyword(s):  
Finite Group ◽  
Free Group ◽  
Group Ring ◽  
Prime Order ◽  
Infinite Order ◽  
Integral Group Ring ◽  
Integral Group ◽  
Cyclic Unit ◽  
Central Elements ◽  
Group A

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

scholarly journals A sufficient condition for the second derived factor group to be finite

1987 ◽  
Vol 30 (1) ◽  
pp. 83-89
Author(s):  
J. R. Howse
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Factor Group ◽  
Sufficient Condition

This paper concerns an application of an algorithm for the second derived factor group as described by Howse and Johnson in [3]. This algorithm has as its basis theFox derivative (see [1]), a mapping from the free group F to the group-ring ℤF, definedas follows: let X be a set of generators of a group G, and let w = y1…yk with each yi∈X±1.

scholarly journals The subgroup determined by a certain ideal in a free group ring

2016 ◽  
Vol 449 ◽  
pp. 400-407 ◽  
Author(s):  
Roman Mikhailov ◽  
Inder Bir S. Passi
Keyword(s):  
Free Group ◽  
Group Ring

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.

Free Groups in Subnormal Subgroups and the Residual Nilpotence of the Group of Units of Groups Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 365-370 ◽  
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Free Group ◽  
Group Ring ◽  
Free Groups ◽  
Unit Group ◽  
Subnormal Subgroup ◽  
Group Of Units ◽  
Residual Nilpotence ◽  
Subnormal Subgroups

AbstractLet KG be the group ring of the group G over the field K and U(KG) its unit group. When G is finite we derive conditions which imply that every noncentral subnormal subgroup of U(KG) contains a free group of rank two. We also show that residual nilpotence of U(KG) coincides with nilpotence, this being no longer true if G is infinite.We can answer partially the following question: when is G sub-normal in U(KG)?

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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