Richardson, James Stephen (1976) Group rings with non-zero socle. PhD thesis, University of Warwick.

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Abstract

The aim of this thesis is to investigate the circumstances under which group rings over fields have non-zero socle, i.e. contain minimal one-sided ideals.

After an introductory chapter, we consider the special case o₊ a periodic abelian group and a non-modular field (that is, a field of characteristic prime to the orders of the elements of the group). This special case, and the background material contained in Chapter III, serve as preparation for our principal results, which concern, locally finite groups.

We establish necessary and sufficient conditions on an arbitrary field K and a locally finite group 0 for the group ring KG to contain minimal one-sided ideals: the most important condition is that G should be a Cernikov group. We then examine the structure of KG when these conditions are satisfied. Vie show that KG has a finite series of ideal3 each factor of which i3 a direct sum of quasi-Frobenius rings, and characterize the socle of KG. Me also classify indecomposable KG-modules, and determine (for countable but not necessarily locally finite groups G) necessary and sufficient conditions for all indecomposable KG-modules to be irreducible.

In the final chapter we consider non-locally-finite groups, conjecturing that group rings of such groups never contain minimal one-sided ideals. We establish the truth of this conjecture for several classes of groups, and also consider semiartinian group rings.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Group rings, Ideals (Algebra), Abelian groups, Finite groups
Official Date:	August 1976
Dates:	Date Event August 1976 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hartley, B.
Sponsors:	Association of Commonwealth Universities ; British Council ; University of Virginia ; University of Warwick ; University of Wisconsin
Extent:	viii, 126 leaves
Language:	eng

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