Karabacak, FatihKosan, M. TamerQuynh, T. CongTasdemir, Ozgur2024-06-122024-06-1220220092-78721532-4125https://doi.org/10.1080/00927872.2021.1979026https://hdl.handle.net/20.500.14551/19916A 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 Schroder-Bernstein property; (4) being virtually extending (VC2) is a Morita invariant property; (4) if M circle plus E(M) is a VC2-module where E(-) denotes the injective hull, then M is injective.en10.1080/00927872.2021.1979026info:eu-repo/semantics/closedAccessCo-Hopfian ModuleOsofsky-Smith TheoremSchroder-Bernstein PropertySquare-Free ModuleVirtually Extending ModuleVirtually C2 ModuleVirtually Semisimple ModuleSubmodulesPropertyArticle50311541168Q4WOS:0007097357000012-s2.0-85117469464Q2