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

KARABACAK F., KOŞAN M. T. , Quynh T. C. , Tasdemir O.

COMMUNICATIONS IN ALGEBRA, vol.50, no.3, pp.1154-1168, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 3
  • Publication Date: 2022
  • Doi Number: 10.1080/00927872.2021.1979026
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Civil Engineering Abstracts
  • Page Numbers: pp.1154-1168
  • Keywords: Co-Hopfian module, Osofsky-Smith Theorem, Schroder-Bernstein property, square-free module, virtually extending module, virtually C2 module, virtually semisimple module, SUBMODULES, PROPERTY
  • Gazi University Affiliated: Yes


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 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.