# Regularity and Substructures of Hom [electronic resource] / by Friedrich Kasch, Adolf Mader.

##### By: Kasch, Friedrich [author.]

##### Contributor(s): Mader, Adolf [author.] | SpringerLink (Online service)

Material type: TextSeries: Frontiers in Mathematics: Publisher: Basel : Birkhäuser Basel, 2009Description: XV, 164 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783764399900Subject(s): Mathematics | Algebra | Associative rings | Rings (Algebra) | Group theory | Mathematics | Associative Rings and Algebras | Group Theory and Generalizations | AlgebraAdditional physical formats: Printed edition:: No titleDDC classification: 512.46 LOC classification: QA251.5Online resources: Click here to access onlineItem type | Current location | Collection | Call number | Status | Date due | Barcode | Item holds |
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Notation and Background -- Regular Homomorphisms -- Indecomposable Modules -- Regularity in Modules -- Regularity in HomR(A, M) as a One-sided Module -- Relative Regularity: U-Regularity and Semiregularity -- Reg(A, M) and Other Substructures of Hom -- Regularity in Homomorphism Groups of Abelian Groups -- Regularity in Categories.

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.

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