This book generalizes the well-known regularity of ring elements to regularity of homomorphisms in module categories. It characterizes regular homomorphisms in terms of decompositions of domain and codomain, and presents numerous other results.

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.

Readable text with new concepts opening new avenues for research Old and numerous new results in self-contained form Results never published in book form Extension of the well-known and important concept of regularity Includes supplementary material: sn.pub/extras

Inhalt
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.