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.

scholarly journals SIMPLE REGULAR SKEW GROUP RINGS

2005 ◽  
Vol 04 (02) ◽  
pp. 127-137 ◽  
Author(s):  
KATHI CROW
Keyword(s):  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Skew Group Rings ◽  
Necessary And Sufficient ◽  
Skew Group Ring

Given a group G acting on a ring R via α:G → Aut (R), one can construct the skew group ring R*αG. Skew group rings have been studied in depth, but necessary and sufficient conditions for the simplicity of a general skew group ring are not known. In this paper, such conditions are given for certain types of skew group rings, with an emphasis on Von Neumann regular skew group rings. Next the results of the first section are used to construct a class of simple skew group rings. In particular, we obtain a more efficient proof of the simplicity of a certain ring constructed by J. Trlifaj.

scholarly journals Regular skew group rings

1995 ◽  
Vol 58 (2) ◽  
pp. 167-182 ◽  
Author(s):  
Ricardo Alfaro ◽  
Pere Ara ◽  
Angel Del Río
Keyword(s):  
Group Ring ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Rank Function ◽  
Necessary Condition ◽  
Group Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Necessary And Sufficient ◽  
Skew Group Ring

AbstractLet G be a group acting on a ring R. We study the problem of determining necessary and sufficient conditions in order that the skew group ring RG be von Neumann regular. Complete characterizations are given in some particular situations, including the case where all idempotents of R are central. For a regular ring R admitting a G-invariant pseudo-rank function N, with G finite, we obtain a necessary condition for RG being regular in terms of the induced action of G on the N-completion of R.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

Extending the Property of a Maximal Right Ideal

2006 ◽  
Vol 13 (01) ◽  
pp. 163-172 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Dinh Van Huynh ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
The Self ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

We extend various properties from a direct summand X of a module M, whose complement is semisimple, to its trace in M or to M itself. The case when MR = RR and the properties are injectivity or P-injectivity is fully described. As applications, we extend some known results for right HI-rings and give a new characterization of semisimple rings. We conclude this paper by giving some conditions that yield the self-injectivity of von Neumann regular rings.

scholarly journals Onn-flat modules andn-Von Neumann regular rings

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Najib Mahdou
Keyword(s):  
Regular Ring ◽  
Local Rings ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Flat Modules ◽  
Von Neumann Regular ◽  
Regular Rings

We show that eachR-module isn-flat (resp., weaklyn-flat) if and only ifRis an(n,n−1)-ring (resp., a weakly(n,n−1)-ring). We also give a new characterization ofn-Von Neumann regular rings and a characterization of weakn-Von Neumann regular rings for (CH)-rings and for local rings. Finally, we show that in a class of principal rings and a class of local Gaussian rings, a weakn-Von Neumann regular ring is a (CH)-ring.

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