The R-module M is said to be
supplemented if every submodule of M has a minimal supplement. For R a Dedekind
domain, we relate this lattice theoretical condition to direct decompositions of M,
the smallness of the radical J(M) of M, the semi-simplicity and lifting of
decompositions of M∕J(M), and the existence of quasi-projective covers. If M
is contained in some R-module as a small submodule, M is said to be a
small module. The structure of all supplemented and all small R-modules is
determined and it is shown that, for R local, the smallness of J(M), the
smallness of M, and M being a supplemented reduced module are equivalent
conditions.
