: The notion of weakly co-Hopfian module is generalized in this way: A module M is called quasi co-Hopfian if M/f(M) is singular, for every injective endomorphism f of M. These modules are studied extensively. Over a right nonsingular ring several equivalent conditions to being quasi co-Hopfian are given. A ring R is semisimple if and only if every quasi co-Hopfian module is co-Hopfian. A ring R is right nonsingular if and only if every R-module of finite reduced rank is quasi co-Hopfian. Every module contains a unique largest fully invariant submodule which is quasi co-Hopfian. This submodule for some modules such as semisimple modules is characterized. Moreover, a quasi-injective weakly co-Hopfian module over a right noetherian ring is characterized. Modules for which every submodule is weakly co-Hopfian (resp. quasi co-Hopfian) is called completely weakly co-Hopfian (resp. completely quasi co-Hopfian). Modules of finite uniform dimension (resp. of finite reduced rank) are completely weakly co-Hopfian (resp. completely quasi co-Hopfian). Those right semi-artinian rings and right FBN rings over which completely weakly co-Hopfian (resp. completely quasi co-Hopfian) modules are precisely modules of finite uniform dimension (resp. of finite reduced rank), are characterized. A submodule A of a module M is called t-essential if A?B is not contained in Z2(M), for every submodule B which is not contained in Z2(M). A module M over a right nonsingular ring is quasi co-Hopfian if and only if f(M) is t-essential in M, for every injective endomorphism f of M. Every essential submodule is t-essential. There is a one to one correspondence between essential submodules of M/Z2(M) and t-essential submodules of M which contain Z2(M). A theory for reduced rank of a module is created in which t-essential submodules have a role similar to the role of essential submodules in the theory of uniform dimension. Tow notions t-complement and t-closed are introduced and it is shown that they are equivalent. A module M is of finite reduced rank if and only if M has ACC on t-complements if and only if M has DCC on t-complements. Finally it is shown that for every R-module M, Z2(M) is the set of those x in M for which ann(x) is a right t-essential ideal of R.