Quasi Co-Hopfian Modules and Applications

We carry out an extensive study of modules M _R with the property that M/f(M) is singular for all injective endomorphisms f of M. Such modules called “quasi co-Hopfian”, generalize co-Hopfian modules. It is shown that a ring R is semisimple if and only if every quasi co-Hopfian R-module is co-Hopfian. Every module contains a unique largest fully invariant quasi co-Hopfian submodule. This submodule is determined for some modules including the semisimple ones. Over right nonsingular rings several equivalent conditions to being quasi co-Hopfian are given. Modules with all submodules quasi co-Hopfian are called “completely quasi co-Hopfian” (cqcH). Over right nonsingular rings and over certain right Noetherian rings, it is proved that every finite reduced rank module is cqcH. For a right nonsingular ring which is right semi-Artinian (resp. right FBN) the class of cqcH modules is the same as the class of finite reduced rank modules if and only if there are only finitely many isomorphism classes of nonsingular R-modules which are simple (resp. indecomposable injective).