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

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 exten...

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Published in:Communications in algebra Vol. 50; no. 3; pp. 1154 - 1168
Main Authors: Karabacak, Fatih, Koşan, M. Tamer, Quynh, T. Cong, Taşdemir, Özgür
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 04.03.2022
Taylor & Francis Ltd
Subjects:
Co-Hopfian module
Modules
Osofsky-Smith Theorem
Schröder-Bernstein property
square-free module
virtually C2 module
virtually extending module
virtually semisimple module
ISSN:0092-7872, 1532-4125
Online Access:Get full text
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Summary: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) if is a VC2-module where denotes the injective hull, then M is injective. Communicated by Scott Chapman
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ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2021.1979026