On some classes of modules and their endomorphism rings

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On some classes of modules and their endomorphism rings

Title: On some classes of modules and their endomorphism rings
Author: Bien, Mai Hoang
Publisher: Mathematical Institute, Faculty of Scoence, Leiden University
Issue Date: 2014-05-27
Keywords: Injective
Max module
Endomorphism ring
Division algebra
Abstract: In [33], it was proved that an injective module is indecomposable if and only if its endomorphism ring is local. The first aim of this thesis is to generalize this result from indecomposable modules to square-free ones by showing that an injective module is square-free if and only if its endomorphism ring is quasi-duo. We then describe all maximal right (left, two-sided) ideals of the endomorphism ring of an arbitrary injective module. We then study two classes of modules: Loewy modules with finite Loewy invariants over an arbitrary ring and max modules with finite radical invariants over a semilocal ring. We prove that the endomorphism rings of these modules are semilocal, generalizing in this way a result that proved by Camps and Dicks for artinian modules [9]. In [12], Facchini proved that, over a commutative ring, a module is artinian if and only if it is a Loewy module with finite Loewy invariants. Here we show that this is not necessarily true for modules over non-commutative rings
Description: Promotores: Hendrik W. Lenstra, Alberto Facchini
With summaries in Dutch and Italian
Faculty: Faculteit der Wiskunde en Natuurwetenschappen
Citation: Bien, M.H., 2014, Doctoral thesis, Leiden University
Sponsor: Algant-Doc Erasmus mundus Program
Handle: http://hdl.handle.net/1887/25834

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