Extending modules are generalizations of injective modules and, dually,
lifting modules generalize projective supplemented modules. There is a
certain asymmetry in this duality. While the theory of extending modules
is well documented in monographs and text books, the purpose of our
monograph is to provide a thorough study of supplements and projectivity
conditions needed to investigate classes of modules related to lifting
modules.
The text begins with an introduction to small submodules, the radical,
variations on projectivity, and hollow dimension. The subsequent
chapters consider preradicals and torsion theories (in particular
related to small modules), decompositions of modules (including the
exchange property and local semi-T-nilpotency), supplements in modules
(with specific emphasis on semilocal endomorphism rings), finishing with
a long chapter on lifting modules, leading up their use in the theory of
perfect rings, Harada rings, and
quasi-Frobenius rings.
Most of the material in the monograph appears in book form for the first
time. The main text is augmented by a plentiful supply of exercises
together with comments on further related material and on how the theory
has evolved.