Regularity and Substructures of Hom (eBook)

Friedrich Kasch, Adolf Mader (Autoren)

eBook Download: PDF

2009 | 2009
XV, 164 Seiten
Springer Basel (Verlag)
978-3-7643-9990-0 (ISBN)

Infos

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.

Inhalt

Hom of 4
Contents 6
Preface 8
Notation and Background 17
Regular Homomorphisms 26
Indecomposable Modules 56
Regularity in Modules 64
Regularity in HomR(A, M) as a One-sided Module 73
Relative Regularity: U-Regularity and Semiregularity 82
Reg(A, M) and Other Substructures of Hom 93
Regularity in Homomorphism Groups of Abelian Groups 115
Regularity in Categories 142
Bibliography 168
Index 171

Produkt-Details

• Erscheint lt. Verlag: 3.3.2009
• Reihe/Serie: Frontiers in Mathematics
• Reihe/Serie: Frontiers in Mathematics
• Zusatzinfo: XV, 164 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: abelian group • Algebra • domain decomposition • Homomorphism • module category • regular homomorphism
• ISBN-10: 3-7643-9990-2 / 3764399902
• ISBN-13: 978-3-7643-9990-0 / 9783764399900