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Irreducible modules and their injective hulls over group rings
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Musson, Malcolm Ian (1979) Irreducible modules and their injective hulls over group rings. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b1751132~S15
Abstract
If R is a ring and V and R module, then there in a unique minimal injective module, containing The module ER(V) is called the injective hull of V. Our aim in this thesis is to study the injective hull of irreducible nodules over various croup rings.
Chapter 1 contains some preliminary results which are used in later chapters. In chapter 2 we study the group algebra of a locally finite group G over a field k. A module E is said to be ∑injective if any direct sum of copies of E is injective. We characterize ∑injective kG modules and provide necessary and sufficient conditions for the injective hull of every irreducible kG module to be ∑injective.
The remaining three chapters concern group rings, SG of polycyclic groups. If R is a commutative Noetherian ring, and V an irreducible R module, it is known that ER (V) is artinian. In chapters 3 and 4, we study analogues of this result. Chapter 3 covers all the cases where we know that ESG (V) is artinian, while in chapter 4 we examine situations in which ESG (V) is not artinian.
In fact we show that ESG (V) can fail to be locally artinian. This answers a question of Jategaonkar concerning (two sided) Noetherian rings.
The main result of chapter 5 concerns irreducible modules over polycyclic group algebras, kG. We shall show that any polycyclicbyfinite group G has a characteristic abelianbyfinite subgroup A, known as the plinth socle of G, such that, if V is an irreducible kG module, the restriction of V to A is generated by finite dimensional kA modules. The motivation is partly a theorem of P. Hall to which the above result reduces when 0 is nilpotent.
Also the condition arose quite naturally in chapter 4, and some applications are given to problems studied there.
A detailed introduction is given separately for each of the chapters 25
Item Type:  Thesis or Dissertation (PhD)  

Subjects:  Q Science > QA Mathematics  
Library of Congress Subject Headings (LCSH):  Injective modules (Algebra), Rings (Algebra), Modules (Algebra)  
Official Date:  July 1979  
Dates: 


Institution:  University of Warwick  
Theses Department:  Mathematics Institute  
Thesis Type:  PhD  
Publication Status:  Unpublished  
Supervisor(s)/Advisor:  Hartley, B., Brown, Kenneth A. (Kenneth Alexander), 1951  
Sponsors:  Science Research Council (Great Britain)  
Extent:  97 leaves  
Language:  eng 
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