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

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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 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

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Injective modules (Algebra), Rings (Algebra), Modules (Algebra)
Official Date:	July 1979
Dates:	Date Event July 1979 Submitted
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Publication Status:	Unpublished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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