Regularity and Substructures of Hom

Adolf Mader

ISBN 10: 3764399899 / ISBN 13: 9783764399894

Editore: Birkhäuser Basel, Birkhäuser Basel Jan 2009, 2009

Riguardo questo articolo

Neuware -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.Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, 14197 Berlin 184 pp. Englisch.

Codice articolo 9783764399894

Dati bibliografici

• Titolo: Regularity and Substructures of Hom
• Editore: Birkhäuser Basel, Birkhäuser Basel Jan 2009
• Data di pubblicazione: 2009
• Lingua: Inglese
• ISBN 10: 3764399899
• ISBN 13: 9783764399894
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 242 L x 170 A x 11 P (millimetri) 242 millimetri di larghezza per 170 millimetri di altezza per 11 millimetri di profondità
• Peso: 326 grammi

Descrizione articolo

Riassunto

The book generalizes the well-known regularity of ring elements to regularity of homomorphisms in module categories, and further to regularity of morphisms in any category. Regular homomorphisms are characterized in terms of decompositions of domain and codomain, and numerous other results are presented. While the theory is well developed in module categories many questions remain about generalizations and extensions to other categories. The book only requires the knowledge of a basic course in modern algebra. It is written clearly, with great detail, and is accessible to students and researchers alike.

Recensione

From the reviews:

“This book is dedicated to generalizations of regularity for an Abelian group ... . contains an excellent and detailed exposition of results on all types of regularity in Hom with consequences for modules and rings. It is accessible, with all necessary definitions and proofs, contains also a series of instructive examples. ... interest both for students and specialists.” (A. I. Kashu, Zentralblatt MATH, Vol. 1169, 2009)