When Some Complement of a Submodule Is a Summand

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 ⊕ M 2, where M has C 11 and M 1 is a fully invariant submodule of M, then both M 1 and M 2 are C 11-modules. Moreover, the C 11 condition is shown to be closed under formation of the ring of column finite matrices of size Γ, 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 from 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.