A module M is said to satisfy the C-11 condition if every submodule of M has a (i.e., at least one) complement which is a direct summand. It is known that the C-1 condition implies the C-11 condition and that the class of C-11-modules is closed under direct sums but not under direct summands. We show that if M = M-1 circle plus M-2, where M has C-11 and M-1 is a fully invariant sulmodule of M-1 then both M-1 and M-2 are C-11-modules. Moreover, the C-11 condition is shown to he closed under formation of the ring of column finite matrices of size Gamma, the ring of m-by-m upper triangular matrices and right essential overrings. For a module M, we also show that all essential extensions of M satisfying C-11 are essential extensions of C-11-modules constructed front M and certain subsets of idempotent elements of the ring of endomorphisms of the injective hull of M. Finally, we prove that if M is a C-11-module, then so is its rational hull. Examples are provided to illustrate and delimit the theory.