A module is extending (or a CS-module, or a module with ?C1?? if every submodule is essential in a direct summand (equivalently, if every closed submodule is a direct summand). Extending modules generalize injective and quasi-continuous ones; they have been studied extensively but many questions remain unresolved. The most annoying open problem is to find sensible necessary and sufficient conditions for the direct sum of extending modules to be extending. Mutual injectivity is sufficient ([1], Theorem 8) but not necessary (as the abelian group