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A right  R -module  M  is virtually extending (or CIS) if every complement submodule of  M  is isomorphic to a direct summand of  M , and  M  is called a virtually C2-module if every complement submodule of  M  which is isomorphic to a direct summand of  M  is itself a direct summand. The class of virtually extending modules (respectively, virtually C2-modules) is a strict and simultaneous generalization of extending modules (respectively, unifies extending modules and C2-modules):  M  is a semisimple module if and only if  M  is virtually semisimple and C2, and  M  is an extending module if and only if  M  is virtually extending and virtually C2. Furthermore, every virtually simple right  R -module is injective if and only if  R  is a right V-ring and the class of virtually simple right  R -modules coincides with the class of simple right  R -modules. Among other results, we show that (1) if all cyclic sub-factors of a cyclic weakly co-Hopfian right  R -module  M  are virtually extending, then  M  is a finite direct sum of uniform submodules; (2) every distributive virtually extending module over any Noetherian ring is a direct sum of uniform submodules; (3) over a right Noetherian ring, every virtually extending module satisfies the Schröder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) ifis a VC2-module wheredenotes the injective hull, then  M  is injective. Communicated by Scott Chapman

On modules and rings in which complements are isomorphic to direct summands