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Title: Irreducible modules and their injective hulls over group rings
Author: Musson, Ian Malcolm
ISNI:       0000 0001 3437 4383
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1979
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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 polycyclic-by-finite group G has a characteristic abelian-by-finite 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 2-5.
Supervisor: Not available Sponsor: Science Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics