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This is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization. The main idea behind the book is to study modules G over a ring R via their endomorphism ring EndR(G). The author discusses a wealth of results that classify G and EndR(G) via numerous properties, and in particular results from point set topology are used to provide a complete characterization of the direct sum decomposition properties of G. For graduate students this is a useful introduction, while the more experienced mathematician will discover that the book contains results that are not otherwise available. Each chapter contains a list of exercises and problems for future research, which provide a springboard for students entering modern professional mathematics.

Endomorphism rings --- Modules (Algebra) --- Endomorphism rings. --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Rings, Endomorphism --- Associative rings

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Ordered algebraic structures --- Categories (Mathematics) --- Endomorphism rings --- Modules (Algebra) --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Rings, Endomorphism --- Associative rings --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Endomorphism rings. --- Anneaux d'endomorphismes. --- Modules (algèbre) --- Catégories (mathématiques)

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Regular rings were originally introduced by John von Neumann to clarify aspects of operator algebras ([33], [34], [9]). A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not ?nite-dimensional ([8, page 155], [32, page V]). Von Neumann proved ([32, Theorem 14. 1, page 208], [8, page 162]): Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. The book of K. R. Goodearl ([14]) gives an extensive account of various types of regular rings and there exist several papers studying modules over regular rings ([27], [31], [15]). In abelian group theory the interest lay in determining those groups whose endomorphism rings were regular or had related properties ([11, Section 112], [29], [30], [12], [13], [24]). An interesting feature was introduced by Brown and McCoy ([4]) who showed that every ring contains a unique largest ideal, all of whose elements are regular elements of the ring. In all these studies it was clear that regularity was intimately related to direct sum decompositions. Ware and Zelmanowitz ([35], [37]) de?ned regularity in modules and studied the structure of regular modules. Nicholson ([26]) generalized the notion and theory of regular modules. In this purely algebraic monograph we study a generalization of regularity to the homomorphism group of two modules which was introduced by the ?rst author ([19]). Little background is needed and the text is accessible to students with an exposure to standard modern algebra. In the following, Risaringwith1,and A, M are right unital R-modules.

Endomorphism rings. --- Homomorphisms (Mathematics). --- Modules (Algebra). --- Rings (Algebra). --- Modules (Algebra) --- Homomorphisms (Mathematics) --- Rings (Algebra) --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Finite number systems --- Modular systems (Algebra) --- Mathematics. --- Algebra. --- Associative rings. --- Group theory. --- Associative Rings and Algebras. --- Group Theory and Generalizations. --- Algebraic fields --- Functions --- Finite groups --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematical analysis