On Semi-Perfect Group Rings

1969 ◽  
Vol 12 (5) ◽  
pp. 645-652 ◽  
Author(s):  
W.D. Burgess
Keyword(s):  
Group Ring ◽  
Jacobson Radical ◽  
Group Rings ◽  
Perfect Group ◽  
Completely Reducible

In what follows the notation and terminology of [7] are used and all rings are assumed to have a unity element.The purpose of this note is to give some partial answers to the question: under which conditions on a ring A and a group G is the group ring AG semi-perfect?For the convenience of the reader a few definitions and results will be reviewed. A ring R is called semi-perfect if R/RadR (Jacobson radical) is completely reducible and idempotents can be lifted modulo RadR (i.e., if x is an idempotent of R/RadR there is an idempotent e of R so that e + RadR = x).

Some Results on Semi-Perfect Group Rings

1974 ◽  
Vol 26 (1) ◽  
pp. 121-129 ◽  
Author(s):  
S. M. Woods
Keyword(s):  
Group Ring ◽  
Polynomial Ring ◽  
Strong Interaction ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Commutative Case ◽  
Perfect Group ◽  
Case Examples ◽  
Complete Answer ◽  
Necessary And Sufficient

The aim of this paper is to find necessary and sufficient conditions on a group G and a ring A for the group ring AG to be semi-perfect. A complete answer is given in the commutative case, in terms of the polynomial ring A[X] (Theorem 5.8). In the general case examples are given which indicate a very strong interaction between the properties of A and those of G. Partial answers to the question are given in Theorem 3.2, Proposition 4.2 and Corollary 4.3.

Rings of Quotients of Group Rings

1969 ◽  
Vol 21 ◽  
pp. 865-875 ◽  
Author(s):  
W. D. Burgess
Keyword(s):  
Group Ring ◽  
Group Rings ◽  
Maximum Condition ◽  
Regular Group ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Completely Reducible ◽  
Rings Of Quotients ◽  
Prime Case

The group ring AG of a group G and a ring A is the ring of all formal sums Σg∈G agg with ag ∈ A and with only finitely many non-zero ag. Elements of A are assumed to commute with the elements of G. In (2), Connell characterized or completed the characterization of Artinian, completely reducible and (von Neumann) regular group rings ((2) also contains many other basic results). In (3, Appendix 3) Connell used a theorem of Passman (6) to characterize semi-prime group rings. Following in the spirit of these investigations, this paper deals with the complete ring of (right) quotients Q(AG) of the group ring AG. It is hoped that the methods used and the results given may be useful in characterizing group rings with maximum condition on right annihilators and complements, at least in the semi-prime case.

On Group Rings

1970 ◽  
Vol 22 (2) ◽  
pp. 249-254 ◽  
Author(s):  
D. B. Coleman
Keyword(s):  
Nilpotent Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Characteristic Zero ◽  
Sylow Subgroup ◽  
Jacobson Radical ◽  
Closed Field ◽  
Group Rings ◽  
Locally Finite ◽  
Image Position

Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

Uniqueness of the Coefficient Ring in Some Group Rings

1973 ◽  
Vol 16 (4) ◽  
pp. 551-555 ◽  
Author(s):  
M. Parmenter ◽  
S. Sehgal
Keyword(s):  
Cyclic Group ◽  
Group Ring ◽  
Jacobson Radical ◽  
Group Rings ◽  
Infinite Cyclic Group ◽  
Coefficient Ring

Let 〈x〉 be an infinite cyclic group and Ri〈x〉 its group ring over a ring (with identity) Ri, for i = l and 2. Let J(Ri) be the Jacobson radical of Ri. In this note we study the question of whether or not R1〈x〉≃R2〈x〉 implies R1≃R2. We prove that this is so if Zi the centre of Ri is semi-perfect and J(Zi〈x〉) = J(Zi〈)x〉 for i = l and 2. In particular, when Zi is perfect the second condition is satisfied and the isomorphism of group rings Ri〈x〉 implies the isomorphism of Ri.

scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

scholarly journals The number of idempotents in abelian group rings

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 719-723
Author(s):  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Group Rings ◽  
Arbitrary Characteristic

Suppose that R is a commutative unitary ring of arbitrary characteristic and G is a multiplicative abelian group. Our main theorem completely determines the cardinality of the set id(RG), consisting of all idempotent elements in the group ring RG. It is explicitly calculated only in terms associated with R, G and their divisions. This result strengthens previous estimates obtained in the literature recently.

TRIVIAL TORSION UNITS IN G-ADAPTED GROUP RINGS

2006 ◽  
Vol 05 (06) ◽  
pp. 781-791
Author(s):  
ALLEN HERMAN ◽  
YUANLIN LI
Keyword(s):  
Group Ring ◽  
Torsion Group ◽  
Unit Group ◽  
Group Rings ◽  
Quaternion Group ◽  
Torsion Units ◽  
Equivalent Conditions

Let G be a torsion group and let R be a G-adapted ring. In this note we study the question of when the group ring RG has only trivial torsion units. It turns out that the above question is closely related to the question of when the quaternion group ring RQ8 has only trivial torsion units. We first give a ring-theoretic condition on R which determines exactly when the quaternion group ring has only trivial torsion units. Then several equivalent conditions for RG to have only trivial torsion units are provided. We also investigate the hypercenter of the unit group of a G-adapted group ring RG, and show that when R satisfies the torsion trivial involution condition, this hypercenter is not equal to the center if and only if G is a Q*-group.

On Automorphisms of Complete Algebras And The Isomorphism Problem for Modular Group Rings

1990 ◽  
Vol 42 (3) ◽  
pp. 383-394 ◽  
Author(s):  
Frank Röhl
Keyword(s):  
Group Ring ◽  
Modular Group ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Affirmative Answer ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Analogous Question

In [5], Roggenkamp and Scott gave an affirmative answer to the isomorphism problem for integral group rings of finite p-groups G and H, i.e. to the question whether ZG ⥲ ZH implies G ⥲ H (in this case, G is said to be characterized by its integral group ring). Progress on the analogous question with Z replaced by the field Fp of p elements has been very little during the last couple of years; and the most far reaching result in this area in a certain sense - due to Passi and Sehgal, see [8] - may be compared to the integral case, where the group G is of nilpotency class 2.

Sign in / Sign up

Export Citation Format

Share Document