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.
